At the moment I use Zetaboards, Before that we were on Invisionfree - both supported it. Most of the free hosted sites do allow it via the admin CP as well (Jcink, Proboards included in that). A few self-hosted sites do as well (usually those with linkable accounts).
There are thousands of writing roleplay forums on the free hosted forums (just check out RPG-Directory.com once they've finished converting to XF). Various factors work into attracting new members, but account per character has proven through play-by-post admin discussions on RPGD (and other places like that) that APC is more popular and that potential members will overlook APC if they prefer APP, but that for those who will overlook it, hundreds of forum-based roleplayers won't join if it is Account Per Player. We have 200+ awards as well, and some of them are earned in-character as well as for out-of-character actions. I've figured out how Xenforo can support the stuff an admin would give without it being automated so that's no longer an issue, but we do have quite a few things to rework if we do still go ahead with swapping over.