I looked into this before purchasing a Dory and found a friend that was willing but basically wanted a signed document saying I'd be responsible for any repairs needed done by a reputable Dory builder. At the end of the day, I decided to find my own since I just didn't want the responsibility of taking someone elses pride and joy down the river. I put in for a February GC trip for next year but didn't win but if I go, I wanna row my Dory. I've definitely let a bunch of other people row it, but usually for an hour or two on the easy stuff. The previous owner has asked to rent it from me for a trip but at the end of the day it just hasn't worked out and I think ultimately I'd be unwilling to let someone else take it out unsupervised.... especially a stranger.... so it just doesn't work really.
I don't know of any companies that rent them either. I think its just too much risk of high levels of damage and on river repairs take some knowledge and skill. There is probably someone willing... but I don't know anyone.
Plenty of time to build/buy one though. I'd consider one that has been well used so you don't wanna drown yourself when you hit a rock the first time. I ended up getting a beat up old Aluminum commercial Grand Canyon dory and its been a lot of fun. Boat seems huge on land but once you get it on the water it feels like its half the size. Definitely a pretty amazing way to go down the canyon.