Are there any Christian groups or denominations that believe that non-Christians have access to the power of the Holy Spirit?

Addressing some misunderstanding

Speaking from mainstream (Trinitarian) groups, your question contains several implicit misunderstandings leading to the framing of your question:

1. Holy Spirit is a person, thus having mind, will, character, purpose. He is a person having the same essence as the other two persons of the Trinity. Therefore, we can learn about his character and purpose from revelations about God the Father and Jesus Christ. For example:
   • God's character is unfailing love and faithfulness, just but merciful, etc.
   • God's purpose vis a vis humanity is to free us from the slavery of sin to a redeemed life (Romans 6) where we can be more like Jesus Christ (Rom 8:29, 1 Cor 15:48-49, Col 3:10, 1 Cor 11:1, etc.) bearing fruits of love since love is the character of the saved. At least that's the character that we want to be transformed into.
2. Being a person, the Holy Spirit is not a "presence field" where you can try to move yourself to be "in the zone", nor a "reservoir of power" where you can tap into (as you put it: "access the power"). That kind of language distorts the nature of the Holy Spirit as a person. Catholic Bishop Robert Barron in his Word on Fire Show Understanding the Holy Spirit explains the proper way to understand who Holy Spirit is and how He works. Some snippets:
   • 3:47-5:30: Christocentric should not be Christomonist (reduction of everything to the second person of the Trinity), Holy Spirit is the person most available to us: "the divine power bringing to bear the Son upon us. He is the one who interprets the Word, the one who applies to individual heart the power of the cross and the resurrection. In a way the Holy Spirit is the one we are most directly in contact with."
   • 7:40-9:33 : Holy Spirit is not an addendum, but the whole operation, scriptural language on Holy Spirit is the stretch that God sends His Son into all forsakenness in the world, and then the Son returns to the Father He brings with him everyone to the Father. It's symbolized by the sign of the cross. This is what is meant by having a "spiritual life": we have been gathered into the love that connects the Father and the Son.
   • 10:10-11:45 : Holy Spirit vs. the "force" (in Star Wars); actus purus (actualization itself) but when He is conceived as power in Star Wards, it's a contingent reality, which is not God. ens commune ipsum esse ; Holy Spirit shares in the one divine essence, the unconditioned reality we just talked about; so while we talk about power of the Holy Spirit in the derivative sense, "they cannot be less than personal"
   • 11:45-17:08 : Answering the question "What does it mean when we use the term Personhood vis a vis the Trinity?"
3. Corollary #1: We don't control Him, but we open the door for Him to come. Power, grace, gifts, etc. come to us while we are in the mode of waiting, surrendering, expecting. Catholics believe that through sacraments we can also receive grace.
4. Corollary #2: We open the door by willing what God wills for us. His primary character is love since the Bible says that God is love, 1 John 4:7-21. God wants us to love, even commands us to love. Therefore, opening the door means wanting to love God and others. Only by uniting our will with God's will can the Holy Spirit come, since God doesn't force his way in. Forcing one's will upon others is not love.

Answering your question

What would Holy Spirit Do

You ask: "do non-Christians have access to the power of the Holy Spirit?" A better question in line with Christian understanding of the who the Holy Spirit is and how the Holy Spirit works (described above) is:

Does the Holy Spirit come to people who either haven't heard the gospel or are currently practicing another religion?

The big difference with the two framing is that in the latter framing the initiative lies with God: what would God do with them?

The ultimate mainstream Christian answer: we do not know how God works with them since the Bible doesn't explicitly say. But given God's character to want to save all people (1 Tim 2:4-6), it's foreseeable that God works in conjunction with missionary effort, God doesn't just sit idle while the Christians do mission work.

It starts with recognizing a problem

All well meaning people recognize in their conscience that love is good and that we should love. Given their upbringing, culture, and the religions they are born in, they will try their best to love, and find that they are failing to do so. Christians call it recognizing how sinful we are, but non-Christians can frame the condition in their own way. If their non-Christian religions fail to help them, and if they pray to God to help them to love better, it is consistent with the Holy Spirit character that He will do something to arrange salvation for them.

Scenario 1: non-Christians seek the gospel and find it

Under fortuitous circumstances, they will then seek out Christians and are presented with the gospel and the true teaching about love, and how they need to surrender and accept the death of Christ as the gateway for better ability to love (conversion), and then trust the promise of God that the Holy Spirit will not abandon them as long as they want to love (sanctification). Even "wanting to love" itself needs God's grace (saving faith is a gift) so they would surrender and ask the Holy Spirit to "inject" them saving grace so they "want to want to love" resulting in a prayer to Jesus like this:

I give up trying on my own, please renew my will so I can love people that I want to love, like my wife and kids. I accept you as my Lord and Savior. I accept the gift of your death on the cross and I'm willing to crucify my sinful life so you can dwell in my heart enabling me to love with Godly love.

THAT PRAYER is an example of what "accessing the power of the Holy Spirit" means: asking to be given the power to love. We see the fruits produced in our life as the Holy Spirit works within us. We recognize some spiritual gifts (that you ask so many questions about) depending on how the Holy Spirit decides to give us in his wisdom and providence. It doesn't matter whether we adopt the stance of cessationist or continuationist; if we have the right disposition and the Holy Spirit decides to give, the gift will come.

Obviously the gospel has to be understood first, which ideally comes through Bible study with a Trinitarian Christian.

Scenario 2: non-Christians invite God to their heart without the gospel

But what if these well-meaning non-Christians who want to love more don't hear the gospel? Unfortunately, we are not given the revelation about this scenario in the Bible explicitly; there are only hints from which there are a few theories (see the table for the 5 views in my other answer). But we can pray for them so the Holy Spirit will come to them regardless, by Holy Spirit revealing himself to their souls directly. Praying for non-Christians to be saved is certainly Biblical, since again, it's consistent with God's own character to want everyone to be saved (1 Tim 2:4-6). Better yet, this give us the motivation to mobilize our churches to do more with the evangelization effort according to the talents that God has given us.