There is actually no Pasuk at all to wear a Tallis. There is a Pasuk to wear Tzitzis and we do that from the Upsheren, when the chinuch in Torah begins and when we hope he can keep them clean enough.
The only Mekor to wear a Tallis is the Bracha of Lehisatef. There is a Medrash about Hakadosh Baruch Hu showing Moshe Rabbeinu that He is wearing a Tallis, like a Baal Tefilla.
Wearing a Tallis by Davenning is more an Inyan than a Din. It puts the person in a situation of strong focus.
The Arizal said that the main point of the Tallis is covering the head. So much so that he held it is a Bracha Levatala if you don’t cover the head. Since we find in the Gemara that until marriage they didn’t cover their head, this can be a Makor for our long-standing Minhag — on this Inyan of wearing a Tallis.