The FS has actually closed the Fresno Dome campground. However, you can camp right at the trailhead if you like, or there are many spots for dispersed camping right off the road, you will see them as you drive by. It's not very busy up there generally, you will find somewhere to stay, no problem.
If you do want a "real" campground the Chilkoot campground is open, its maybe a ~25 min drive from the climbing. not sure how busy it is, I've never stayed there myself