The Style of Game is typically called Play By Mail or PBM. In the UK, it was also called Play by Post, but given the US domination of the early internet, PBP came to mean posts to forums or BBSs. Play by electronic mail is usually called PBEM.
There are a few games I know that are suitable.
Whether they truly cross all the way into "roleplaying" is arguable, but several of the old play-by-mail games are still run on a commercial basis by Flying Buffalo
En Garde! (originally by GDW, in print by Margam Evans, Ltd.) is likewise on the border between board game and roleplay, and is well suited to play by mail, play by [forum] post, and play by email.
Lords of Space isn't RPGing, but is a human moderated PBM ruleset for play of space empires at war. It was my first encounter with PBM moderator rulesets for sale, tho it actually postdates En Garde. When I ran it, many of my players got heavy into RP-mode.
Fiasco likewise should be playable that way.
Marvel Universe also should be playable by mail, tho' it would be annoyingly slow to me.