Jasper is one of our favourite camping areas.
I would think that in May and June you would have no problem getting into the camp grounds without a reservation with the exception of perhaps maybe the weekends. They have a very large capacity with Whistlers, Wapiti and a few smaller ones.
Although Jasper gets busy in the summer, we have never felt like it is overrun with people.
There are some great bike trails which is one of our must does as well as lots of great hiking trails.
Like LarryandLiz, I am starting to dream as well.