First off great idea. It's one of the best things to see students working with students from other teams. I'm glad you're one of those types of students. It's why I'm in FIRST as opposed to BatteBots.
Pit Admin does have Spare Parts (not sure about tools) so generally the things teams ask for are pretty specific, that even an extremely veteran team may have. I'm not sure as to the specifics of what Spare Parts are there so I don't think this really hampers your idea.
I would think it's just too many tools to organize and it's way easier to let other teams to help with those requests.
I haven't observed enough how quickly it takes to get these requests handled, but perhaps the FIRST Pit Crew (no quotes because it's a real thing now
) could be a group of volunteers that are knowledgeable about teams and know what teams could help out with requests and go seek them out.
Teams are always willing to help but while in the Pit quickly get caught up with their own robot. If someone were to go actually face-to-face ask them then they would jump at lending a tool or a part.
Thoughts?
Tools wise you could always suggest adding some common tools to the Spare Parts station that exists.
Edited Part: Going off of the other two comments, you can develop a network of volunteers that add to the Spare Parts with tools, so that each regional doesn't have to add the tools to their cost. My current home regional is Midwest and I have some wireless drills and some other tools that I would not mind bringing anywhere I go.