Are there any Christian Restorationist denominations that believe their establishment was the fulfillment of Biblical prophecy?
If so, which denominations and which prophecies?
The Church of Jesus Christ of Latter-day Saints, or Mormons, believe their establishment was a fulfillment of Biblical prophecy. Some of the prophecies and their fulfillments:
rod out of the stem of Jesse...And in that day there shall be a root of Jesse, which shall stand for an ensign of the people; to it shall the Gentiles seek...
Joseph Smith fulfills this prophecy of the “root” in two ways. First, the Lord told Joseph that the priesthood rightly belonged to him by lineage (See D&C 86:8-10). Second, Joseph was given the “keys of the kingdom” by a succession of heavenly messengers (See D&C 13:1; D&C 27:5; D&C 110:11–13, 16; D&C 128:20–21). 1
Isaiah 29:11-14 - learned man can't read sealed book, but unlearned man can
to know which of all the sects was right, that I might know which to join, God and Jesus appeared unto him and answered
they draw near to me with their lips, but their hearts are far from me, they teach for doctrines the commandments of men, having a form of godliness, but they deny the power thereof.
I cannot read a sealed book.
Isaiah 40:3-5 - voice crieth in the wilderness
Here, “the voice of him that crieth” does not refer to John the Baptist in the meridian of time; the context of the chapter deals with happenings that will occur just prior to the Second Coming and the Millennium. Therefore, it is likely that Joseph Smith is “the voice of him that crieth.” 1
Ezekiel 37:16-17 - two 'sticks', stick of Judah and stick of Ephraim, joined into one. The LDS believe the stick of Judah is the Holy Bible and the stick of Ephraim is the Book of Mormon
The stick or record of Judah … and the stick or record of Ephraim … are now woven together in such a way that as you pore over one you are drawn to the other; as you learn from one you are enlightened by the other. They are indeed one in our hands. Ezekiel’s prophecy now stands fulfilled.2
Malachi 3:1-3 - prophesy of a forerunner of Christ. See footnote 1, on how this doesn't just apply to Christ's 1st coming, specifically verse 2:
Who may abide the day of his coming? and who shall stand when he appeareth? for he is like a refiner’s fire, and like fullers’ soap
Daniel 2:44 - God will set up a kingdom that shall never be destroyed and that will stand forever
The Church of Jesus Christ of Latter-day Saints was restored in 1830 after numerous revelations from the divine source; and this is the kingdom, set up by the God of heaven, that would never be destroyed nor superseded, and the stone cut out of the mountain without hands that would become a great mountain and would fill the whole earth. 3
Matt 24:14 - gospel preached in all the world before the end comes
Both Laub’s and Richards’s accounts seem to indicate that Joseph taught he was this witness “chosen for the last dispensation or Seventh Dispensation” 1
Revelation 14:6-7 - angel flying in the midst of heaven preaching the everlasting gospel
There is little doubt that John’s vision of “another angel fly[ing] in the midst of heaven” points to the latter days.1, 4
Isaiah 2:2 - mountain of the Lord's house shall be established in the tops of the mountains
This prophecy was fulfilled and further confirmed by Brigham Young, who, when entering the [Salt Lake] valley, rose from his sick bed in Wilford Woodruff’s carriage and said, “This is the right place!”6
Note on Joseph Smith:
If a person thinks the name of Joseph Smith ought to be found in the Bible spelled out in so many letters, he will search in vain.5
5 Answers to Gospel Questions, 3:6, by Joseph Fielding Smith
From the Jehovah's Witnesses publication "Reasoning from the Scriptures"under the heading Jehovah's Witnesses much info about how they view their organization is found.
Key to your question is the following:
How old is the religion of Jehovah’s Witnesses?
According to the Bible, the line of witnesses of Jehovah reaches back to faithful Abel. Hebrews 11:4–12:1 says: “By faith Abel offered God a sacrifice of greater worth than Cain . . . By faith Noah, after being given divine warning of things not yet beheld, showed godly fear . . . By faith Abraham, when he was called, obeyed in going out into a place he was destined to receive as an inheritance . . . By faith Moses, when grown up, refused to be called the son of the daughter of Pharaoh, choosing to be ill-treated with the people of God rather than to have the temporary enjoyment of sin . . . So, then, because we have so great a cloud of witnesses surrounding us, let us also put off every weight and the sin that easily entangles us, and let us run with endurance the race that is set before us.”
With reference to Jesus Christ, the Bible states: “These are the things that the Amen says, the faithful and true witness, the beginning of the creation by God.” Of whom was he a witness? He himself said that he made his Father’s name manifest. He was the foremost witness of Jehovah.—Rev. 3:14; John 17:6.
Interestingly, some of the Jews asked whether the activity of Jesus Christ represented “a new teaching.” (Mark 1:27) Later, some Greeks thought the apostle Paul was introducing a “new teaching.” (Acts 17:19, 20) It was new to the ears of those who were hearing it, but the important thing was that it was the truth, in full harmony with God’s Word.
The modern-day history of Jehovah’s Witnesses began with the forming of a group for Bible study in Allegheny, Pennsylvania, U.S.A., in the early 1870’s. At first they were known only as Bible Students, but in 1931 they adopted the Scriptural name Jehovah’s Witnesses. (Isa. 43:10-12) Their beliefs and practices are not new but are a restoration of first-century Christianity.