Do any Protestant denominations require that their members proselytize?
The denominations I am familiar with don’t keep detailed records about the activities of their members like that, but proselytizing, or witnessing as we Baptists call it, is certainly encouraged. The basis for this is obvious. We have the “good news” that despite our sins, God sent his son to save us and bring us into his kingdom. Within this faith, witnessing is among the most important gifts a believer can share with a non-believer and arguments for “freedom from proselytization” are at best, outside of the faith and at worst, the work of Satan (a.k.a. “the deciever”)
Participation in this scheme is voluntary. Rejection of it would cause concern, perhaps. If you were subject to an overzealous and judgmental congregation, you might feel a good bit of pressure to conform – but then such is the case with any group of believers, religious or libertarian.
Scriptural reference: Mark 16:15-16 “And he said unto them, Go ye into all the world, and preach the gospel to every creature. He that believeth and is baptized shall be saved; but he that believeth not shall be damned.”
(By the way, this passage goes on to indicate the signs by which the followers shall know other believers, which include the snake-handling and the speaking in tongues.)
Every Mormon is a missionary. See here this. Most of the door-to-door proselytizers are young men who volunteer.