Disclaimer: THIS IS MY OPINION, so take it with a grain of NaCl.
FIRST as a whole may not have a lot of money being a non-profit (or the majority of their 50mil/yr budget actually going to teams), but they do have a whole raft of competent technical people willing to help.
If FIRST was to partner with TBA developers, GitHub, Google, or basically anyone with decent knowledge of client-server development, they could either just clean it up in house or make it open-source and have contributions count towards Chairman's
Community Service for all of FIRST teams
might help, or at the very least incentivize it.
The way I'd tackle it regardless is pretty straightforward. Rewrite TIMS / VIMS / STIMS from scratch, and import the existing data via a data dump. They started off as one system (as evidenced by a single piece of paper I found in 1257's file cabinet from 2004) and fragmented it. To clean up the legacy cruft everyone *knows* is there would fix a majority of the problem.
On top of that, not having it written in ASP.NET would be a enormous improvement imho. Nginx and Apache are cheaper to host, plus you don't have to fight with Microsoft lisensing.
I digress. If you don't have a way to register a team, students or volunteers, you don't have them at all. Technology should not be a barrier to entry into a *technology* competition.
Best of luck to all, teams to register students, volunteers to do that awesome thing you do (sacrifice your personal time for robots), and to FIRST if they ever want to move away. It's getting more and more difficult...