"Rapture" is an invented noun to concisely describe some events prophesied in the New Testament, mostly this passage in I Cor. 15:

Listen, I tell you a mystery: We will not all sleep, but we will all be changed— in an instant, in the twinkling of an eye, at the last trumpet. For the trumpet will sound, the dead will be raised imperishable, and we will be changed. For the perishable must be clothed with the imperishable, and the mortal with immortality.…

Also, I Thessalonians 4:

For the Lord himself will come down from heaven, with a loud command, with the voice of the archangel and with the trumpet call of God, and the dead in Christ will rise first. After that, we who are still alive and are left will be caught up together with them in the clouds to meet the Lord in the air. And so we will be with the Lord forever.

Even this does not really describe the rapture as conceived by evangelical Christians, there are a lot of florid passages like this that get taken (and embellished) different ways by different sects. There are for example pre-, mid- and post-tribulation beliefs for what this means ... people who believe in "the rapture" are pre-tribulationists who believe True Christians[tm] will be caught up into heaven and escape the Great Tribulation. Which itself is a subject of much confusion, as there are people who believe it is well in the past already (the sacking of Jerusalem by Titus), others who believe it in the future, and everything in between.

Christian eschatology is every bit the fever dream you'd expect it to be from reading The Revelation.

The word rapture is not in the Bible[en.wikipedia.org]
Nor is the trinity, thevirgin birth was added by Jerome when he wrote the Latin vulgate in 382 C.E. under duress,[biblicalunitarian.com] the Virgin birth was a mistranslation from Hebrew into Greek fom young woman to virgin[patheos.com] There are many things that are believed by Christians that are simply not factual.

If you mean evidence, then there is none.

In the book "Fantasyland" the author (I can't remember his name, it's late) talks about an evangelical group came up with it. Sorry I can't come up with more right now.