Would Mormons accept communion, administered by a representative of another Christian denomination, as valid?

It is a fundamental doctrine of the Latter Day Saints (Mormons) that all other churches are 'corrupt', by which they mean no longer teaching the truth about God or following his commands. As such other churches have no authority with regard to sacrements or anything else, nor any effectiveness regarding sacrements. Taking sacrements from other clergy would not be seen as any kind of benefit. That's not to say that nobody did it, but not with the sanction of the church.

As a member of the church of Jesus Christ of latter day saints, or better known as "mormons", here is the correct answer. And I say correct because you can confirm this with any member of the church, from a 16 yr old priest to the president of the church.

We cannot participate in the communion if it's administered by a person from any other religion, and the reason for that is authority.

When Jesus Christ restored his church through the prophet Joseph Smith, other very important aspect restored was his authority, the priesthood, this happened when John the Baptist as well as Peter, James, and John appeared as resurrected beings to Joseph Smith and Oliver Cowdery.

John the Baptist restored the Aaronic priesthood, and with this, the authority to perform baptisms and bless the bread and the water.

The three Apostles of Jesus Christ layed their hands upon their heads and conferred the Melchisedeq priesthood and with this, the authority to perform the confirmation and other necessary ordinances performed in the temple.

Whith all this, now the authority of God to act in his name for the salvation of his children was again upon the earth.

Then Joseph Smith and Oliver Cowdery in turn conferred this authority to others and these others to some more through generations until this day thus having a chain of authority from Jesus Christ to his original 12 Apostles and from 3 of the Apostles to Joseph Smith, and from Joseph Smith to other people and so forth.

So, any worthy member of the Church of Jesus Christ of latter day saints that possesses this authority, the Aaronic priesthood and has been ordained in the office of "priest", can bless the water and the bread for other members to partake.

But, no other person in any other religion possesses this authority, because to have it, they should have been baptized and confirmed with the proper authority as restored by Jesus Christ.

There are several ways to look at this question.

A. Does the LDS Church believe the Sacrament (the "sacrament of communion" or the Eucharist) when administered by another church or Christian tradition is authoritative?

No. We do not.

We believe that "a man must be called of God, by prophecy, and by the laying on of hands by those who are in authority, to preach the Gospel and administer in the ordinances thereof." (LDS AoF #5). Without that authority, the Sacrament is not acceptable before the Lord (Jer 6:18-20).¹ This belief is supported by the Apostle Paul:

Let every soul be subject unto the higher powers. For there is no power but of God: the powers that be are ordained of God. Whosoever therefore resisteth the power, resisteth the ordinance of God: and they that resist shall receive to themselves damnation.² (Rom 13:1-2, see also Heb 5:1-4)

We believe John the Baptist had the proper authority by lineage to be a priest within the Aaronic Priesthood:

Son of Zacharias and Elisabeth, being of priestly descent through both parents. This lineage was essential, since John was the embodiment of the law of Moses, designed to prepare the way for the Messiah and make ready a people to receive Him. He was the outstanding bearer of the Aaronic Priesthood in all history and was entrusted with its most noble mission. (Source, see also D&C 84:28)

Therefore, we do not accept the authenticity of ordinances by anyone not ordained by those who have the authority to the Aaronic or Melchizedek priesthoods.

But that's very different than...

B. Does the LDS Church teach its members to refuse participation in the Sacrament (the "sacrament of communion" or the Eucharist) when administered by another church or Christioan tradition?

There is no Church teaching that I am aware of, or can find in rapid research, that teaches that a member of The Church of Jesus Christ of Latter-day Saints should not accept a sacramental offering (from the LDS perspective) administered by an officiator of another church.

The idea that we would act with such incivility and rudness is (or, at least, should be) anathema to us. We delight in every opportunity to teach others about our faith, but we should never persecute another for their faith. The Lord taught:

And whosoever shall offend one of these little ones that believe in me, it is better for him that a millstone were hanged about his neck, and he were cast into the sea. (Mark 9:42)

And again:

And Jesus said unto him, Forbid him not: for he that is not against us is for us. (Luke 9:50)

Regrettably, those ideals are applicable to the Church and its membership generally and not individually. We do not believe that people are perfect or saved simply because they join the Church, and there are members who behave less than admirably.

Nevertheless, there are reasons why it would be appropriate and valuable for an LDS member to support the solidarity of a group in crisis. What value could there be in creating ill will, dissent, or anger when the next moment a battle will begin or the group must steel itself before rescue? The Lord taught that we should agree with our adversaries quickly, lest the resulting disunity³ do more harm than good (Matt 5:25) And He ultimately taught that we should love even our enemies, blessing those who hate our Church, and praying for those who would persecute us. (Matt 5:44)

Conclusion

There is nothing that I know of, nor anything that I believe, that would suggest an LDS member should not participate in the Sacrament as provided by another officiator in the circumstances you suggest. On the contrary, the teachings of Jesus Christ are that we should be supportive and consider the needs of the many before our own selfish desires,⁴ always acting within the law that Christ has taught us,⁵ that our good example may grant us opportunity to teach Christ all the more. (1 Cor 9:19-22)

(While I have not been in a crisis situation, I have been invited by friends to attend their church services, where I politely participated in their ordinances and suspect the Lord was delighted with their effort to grow closer to Him and my willingness to let a sleeping dog lie for a greater good. Thus, I do not believe I was in error in any way and the courtesy was returned when they visited my church.)

And though it is true that we do not accept the officiator's authority, we should never act to bring anger, hate, or distrust among a group that must depend on one another (or ever, frankly).

I am grateful for a simple quote from Victor Hugo.

Certain thoughts are prayers. There are moments when, whatever be the attitude of the body, the soul is on its knees.

There really are no atheists in foxholes. When the pressure's on, all men seek for salvation.

¹ That verse from Jeremiah, from the perspective of LDS theology, supports the idea that without proper authority the ordinances performed are not acceptable to the Lord. This is due in part to D&C 131:7 which teaches that when any priesthood holder is wicked, "behold, the heavens withdraw themselves; the Spirit of the Lord is grieved; and when it is withdrawn, Amen to the priesthood or the authority of that man." Said another way, to the Lord it doesn't matter if you had the priesthood and are in sin, or didn't have the priesthood at all. The consequence is the same: the ordinance goes unrecognized.

² I'm obviously viewing this verse through the lenses of LDS theology. Whether or not ours is the true authority of God is a discussion not within the scope of this answer.

³ I consider being thrown into prison an obvious sign of disunity and to be more harmful than good both theologically and physically. I'm just sayin'.

⁴ Yup, I'm a Star Trek fan.

⁵ As mentioned by others, we wouldn't drink the wine, but would ask for an alternative.

Life long member of a multi generational family in the LDS Church. I have attended other religious meetings and even attended a First Communion while serving a mission. Without going into a lot of detail here, many of the answers given with the related citations are correct. A devout Mormon that understands the teachings of the Church regarding Priesthood Authority and the Atonement of Christ would not participate in the taking of sacrament or Communion that was administered by someone that wasn't a member of the of the LDS Church. Said member must also be in good standing and ordained to the office of Priest or higher.

If you are member of the LDS church and male there is a high likelihood that you yourself are an ordained Priest and if no one else was available you could administer the sacrament to yourself. My grandfather served in the US Army during WWII. My 2nd oldest son is in the US Marine Corps.

Additionally the the Church Military Relations department has Senior Missionaries (think older retired couples) serving at US Military Bases. All of these missionaries under the Military Relations department are also retired Military and so have experience both in the military and in the LDS Church. The Military Relations department can assist service members or their families in locating other LDS even when aboard a ship or deployed to a remote base.

Now considering some more exceptional situation, such as when serving in U.S. military during a war, as in the book cover shown below, would Mormons accept the sacraments from military chaplains that may not be Mormons?

No.

5 We believe that a man must be called of God, by prophecy, and by the laying on of hands by those who are in authority, to preach the Gospel and administer in the ordinances thereof.

-- The Articles of Faith, emphasis added

It is a fundamental tenet of Mormonism that the church that Christ organized fell into apostasy and had to be restored, and that among the things that were lost was the Priesthood authority which is necessary to perform the ordinances that pertain to the Gospel. Anybody who has not been properly ordained to the restored Priesthood, therefore, cannot perform such ordinances with any validity, according to LDS doctrine.

Or would a Mormon in this case not participate in the sacrament, or have the sacrament on their own (or possibly request a Mormon chaplain)?

On the lds.org website, we can find official practices and instruction for preparation for military service. Certain soldiers are called and set apart, as in other Church callings, to act as "Service Member Group Leaders," who are not military chaplains but are authorized to organize and lead LDS meetings, see to the spiritual needs of other Mormon soldiers, and preside over the administration of ordinances such as the sacrament.