It is the same program regardless of what school you go to. Army ROTC students at College of Charleston participate in the Army ROTC program through The Citadel.
I don't know for certain at the other senior military colleges, but if there are smaller schools nearby, there is a good chance they would be affiliated with the senior colleges.
The real difference is if you don't go into ROTC. Even if you aren't ROTC, you will get a slightly more "military" experience than a non-ROTC student at a regular college/university.
...and, yes, if the school has ROTC or is affiliated with a school that offers it, you can take it.
FIRST, you get accepted into the college, then, you sign up for ROTC. Anyone can take classes the first 2 years. A 4 year scholarship is a bit different because you will be starting that process before acceptance/enrollment in college, but it allows you to go to any college where you are accepted.