What denominations believe that a Book of Acts style of interaction with the Holy Spirit is still possible today?

The Church of Jesus Christ of Latter-day Saints:

6 We believe in the same organization that existed in the Primitive Church, namely, apostles, prophets, pastors, teachers, evangelists, and so forth.

7 We believe in the gift of tongues, prophecy, revelation, visions, healing, interpretation of tongues, and so forth.

...

9 We believe all that God has revealed, all that He does now reveal, and we believe that He will yet reveal many great and important things pertaining to the Kingdom of God. (Articles of Faith 6,7,9)

The Orthodox and Catholic Churches believe this so for them.

History has shown this to be the case. I will give one example, for each century has many examples of your so called “Book of Acts level of connection with the Holy Spirit.”

St. Augustine’s Conversion and his ”Tolle, lege! Tolle, lege!” moment:

“But when a deep consideration had from the secret bottom of my soul drawn together and heaped up all my misery in the sight of my heart; there arose a mighty storm, bringing a mighty shower of tears. Which that I might pour forth wholly, in its natural expressions, I rose from Alypius: solitude was suggested to me as fitter for the business of weeping; so I retired so far that even his presence could not be a burden to me. Thus was it then with me, and he perceived something of it; for something I suppose I had spoken, wherein the tones of my voice appeared choked with weeping, and so had risen up.

He then remained where we were sitting, most extremely astonished. I cast myself down I know not how, under a certain fig-tree, giving full vent to my tears; and the floods of mine eyes gushed out an acceptable sacrifice to Thee. And, not indeed in these words, yet to this purpose, spake I much unto Thee: and Thou, O Lord, how long? how long, Lord, wilt Thou be angry, for ever? Remember not our former iniquities, for I felt that I was held by them. I sent up these sorrowful words: How long, how long, “tomorrow, and tomorrow?” Why not now? why not is there this hour an end to my uncleanness?

So was I speaking and weeping in the most bitter contrition of my heart, when, lo! I heard from a neighbouring house a voice, as of boy or girl, I know not, chanting, and oft repeating. ‘Take up and read; Take up and read.’ [’Tolle, lege! Tolle, lege!’] Instantly, my countenance altered, I began to think most intently whether children were wont in any kind of play to sing such words: nor could I remember ever to have heard the like. So checking the torrent of my tears, I arose; interpreting it to be no other than a command from God to open the book, and read the first chapter I should find…

Eagerly then I returned to the place where Alypius was sitting; for there had I laid the volume of the Apostle when I arose thence. I seized, opened, and in silence read that section on which my eyes first fell: ‘Not in rioting and drunkenness, not in chambering and wantonness, not in strife and envying; but put ye on the Lord Jesus Christ, and make not provision for the flesh, in concupiscence.’ [Romans 13:14-15] No further would I read; nor needed I: for instantly at the end of this sentence, by a light as it were of serenity infused into my heart, all the darkness of doubt vanished away.” - “tolle lege”, take and read…

Besides such examples, St. Paul expresses that the Holy Spirit will be with the faithful in the end times.

In the last days, God says, I will pour out my Spirit on all people. Your sons and daughters will prophesy, your young men will see visions, your old men will dream dreams. - Acts 2:17

Lumen Gentium of Pope St. Paul affirms that the Holy Spirit will always be with the Church on earth.

The Eschatological Nature of the Pilgrim Church and it’s Union within the Church in Heaven

48. The Church, to which we are all called in Christ Jesus, and in which we acquire sanctity through the grace of God, will attain its full perfection only in the glory of heaven, when there will come the time of the restoration of all things. At that time the human race as well as the entire world, which is intimately related to man and attains to its end through him, will be perfectly reestablished in Christ.

Christ, having been lifted up from the earth has drawn all to Himself. Rising from the dead He sent His life-giving Spirit upon His disciples and through Him has established His Body which is the Church as the universal sacrament of salvation. Sitting at the right hand of the Father, He is continually active in the world that He might lead men to the Church and through it join them to Himself and that He might make them partakers of His glorious life by nourishing them with His own Body and Blood. Therefore the promised restoration which we are awaiting has already begun in Christ, is carried forward in the mission of the Holy Spirit and through Him continues in the Church in which we learn the meaning of our terrestrial life through our faith, while we perform with hope in the future the work committed to us in this world by the Father, and thus work out our salvation.

Already the final age of the world has come upon us and the renovation of the world is irrevocably decreed and is already anticipated in some kind of a real way; for the Church already on this earth is signed with a sanctity which is real although imperfect. However, until there shall be new heavens and a new earth in which justice dwells, the pilgrim Church in her sacraments and institutions, which pertain to this present time, has the appearance of this world which is passing and she herself dwells among creatures who groan and travail in pain until now and await the revelation of the sons of God.

Joined with Christ in the Church and signed with the Holy Spirit "who is the pledge of our inheritance", truly we are called and we are sons of God but we have not yet appeared with Christ in glory, in which we shall be like to God, since we shall see Him as He is. And therefore "while we are in the body, we are exiled from the Lord and having the first-fruits of the Spirit we groan within ourselves and we desire to be with Christ"'. By that same charity however, we are urged to live more for Him, who died for us and rose again. We strive therefore to please God in all things and we put on the armor of God, that we may be able to stand against the wiles of the devil and resist in the evil day. Since however we know not the day nor the hour, on Our Lord's advice we must be constantly vigilant so that, having finished the course of our earthly life, we may merit to enter into the marriage feast with Him and to be numbered among the blessed and that we may not be ordered to go into eternal fire like the wicked and slothful servant, into the exterior darkness where "there will be the weeping and the gnashing of teeth". For before we reign with Christ in glory, all of us will be made manifest "before the tribunal of Christ, so that each one may receive what he has won through the body, according to his works, whether good or evil" and at the end of the world "they who have done good shall come forth unto resurrection of life; but those who have done evil unto resurrection of judgment". Reckoning therefore that "the sufferings of the present time are not worthy to be compared with the glory to come that will be revealed in us", strong in faith we look for the "blessed hope and the glorious coming of our great God and Saviour, Jesus Christ who will refashion the body of our lowliness, conforming it to the body of His glory, and who will come "to be glorified in His saints and to be marveled at in all those who have believed".