i don't believe that's true. Eventually, most coaches reward those guys with scholarships, but I don't think it's a must. It's definitley not a must after 1 season. I know of guys who were on a team for two full seasons before getting a scholarship.
Walkons do not have to ever be offered a scholorship. If your team choose to, they could have all non-scholorship players. Hence the IVY league. There are 3 types of players: scholorship, non-scholorship, and walk-on.