You often do get an AAA discount when camping in RV parks (Remember, the second A in AAA is "Automobile".) Since I have that anyway, I use it. Works at motels of course too, and there are discounts for a surprising number of other random things, but most auto-related. One of the most useful things about AAA is you can get free maps and camp books - and the camp book all types of camping - from forest service to RV park.
There is no universal camping pass for all the national or state parks. Some state park systems have an entry pass (Colorado) like the NPS's (National Park Service) pass. But again, that's just entry, not camping.
The NPS pass just covers entry fees, as mentioned, and is only going to be of use to you if you plan to ride through more than about 4 parks, or if you plan to use it in parks the rest of the year while not bike touring. I've had one for years, since I tend to go to a lot of rock climbing places and cycling areas in parks.
The least expensive formal/pay camping is usually NFS (National Forest Service) campgrounds or BLM (Bureau of Land Management), which are fairly common in the western US, and usually cost $6-12. There is no pass for these. BLM and NFS both have web sites, you could look on those to see if there are any map resources.
A lot of western states have a state recreation map that shows locations for the NFS/BLM camping in that state. I don't know a resource where you can get it all at once - but I got my WY and CO ones in gas stations in-state. Detailed Gazette style maps (like Delorme) also often note the camping locations, but this can be tough on the bike, as the locations aren't always clearly indicated - like you might get near there and fine it's 5 miles up some dirt road.
If you plan to camp in RV parks a LOT, some of them have their own specific membership passes (like Good Sam, I think), but I've never bothered with that, because on a bike you just can't pick which park you go to all the time, it's just forced on you by location.
I guess the closest thing to a universal discount card is the AAA membership.