Not FRC specifically, but it's quite common for competition teams to be a senior design project at Rose-Hulman. I'm working on the Shell EcoMarathon competition as mine, and I know that there at least two other groups (Formula SAE, ASME Human Powered Vehicle) doing the same. Here, funding comes from the team budget, and the "client" for the senior design project is typically the faculty adviser of the competition team.
For FRC specific stuff, there is the classic 254/968 gearbox that was a Cal Poly Pomona senior design project.
https://www.chiefdelphi.com/media/photos/27018
https://www.chiefdelphi.com/media/photos/28709
To help answer the question, what facet of mechanical engineering are you aiming towards focusing this project in? Mechatronics, statics/structural engineering, control theory, materials, etc are all fields that would be applicable since Mechanical Engineering is so broad.