Just circling back with a bit of info I found, in case anyone in the future finds it useful.
This year (2022), there are 4 campgrounds open that are first come, first served (FCFS) and have accessible sites. They are:
- Many Glacier
- Bowman Lake (#3, #7)
- Kintla Lake (#13)
- Rising Sun (#7)
- Two Medicine (#20) - has easy access to Running Eagle Falls (one of the few accessible trails in the park)
These are the only places where the accessible sites will be blocked off to the general public. The other campgrounds that require reservations do not block off any of their accessible sites. For the campgrounds listed above, the accessible sites are ONLY blocked off as long as there are other sites available and being assigned. Once those run out, the accessible sites will be opened up to the general public. The ranger at the Many Glacier campground I spoke to said that in the summer months (peak season), people start lining up for the FCFS sites as early as 5am, and they typically fill up before 10am. Wha this means is that assuming there are not any other people in line for 1 of the accessible sites you're after, all you have to do is get there before 10am (or whenever they fill up). Obviously, the later you go, the greater the chance that you WON'T get there before they fill up and the accessible site will be given to the next person in line.
https://www.nps.gov/applications/glac/cgstatus/camping_detail.cfm?cg=Many%20Glacier - this page, when it is online (often produces an error page), allows you to view historical data on what times the Many Glacier campground has filled up by in the past. if you look at, say July 2021, you'll see that they were generally filled up much earlier than 10am. I thought this was extremely helpful.
And finally, I was told by a general park representative (not sure i he was a ranger or what his actual title was) that Two Medicine campground was the easiest to get into, in general. No idea what that was based on, so I'll be taking it with a grain of salt.
I was able to reserve a campsite (cancellation), so it tuns out I won't have to go through this process, but if I get anymore insight into the process while I'm there, I'll post back upon my return.