Dr. Joe,
Autodesk
student licenses are now available to any high school or college student. All that is required is that the the student must sign up online with a school e-mail address, or send a request to a teacher with a school e-mail address to approve them as a student. The Autodesk licenses are one-year licenses of the full program and any files created with it carry a student version watermark when printed out or added to a non student version file.
Pros:
Students can download from home
Licenses are free for students
All of Autodesk's programs are available (AutoCAD, Inventor PRO, Inventor Publisher, Showcase, 3DS Max, etc.)
Cons:
You aren't familiar with Autodesk Inventor
Even though the only CAD program I know is Inventor, I have always found the interface to be a bit cumbersome
In terms of Solidworks, there is a Solidworks rep for FIRST (correct me if I'm wrong but I think her name is Marie Planchard) who you might be able to PM about getting a few Solidworks licenses for your team. The most popular programs for FIRST are definitely Solidworks and Inventor and there will be resources in FIRST everywhere for students to ask questions. Either program is perfectly capable for FIRST.
Hope this helps!
Jack