Well, I wasn't in FIRST, but I have had experience in the area before.
In my Senior year, I had a volunteer internship with the local public TV station, which happened to be located inside the school; the Hamilton Public Television Station, or HTV for short. The internship took up two classes, but as another class I had a pilot course, Advanced Study In Filmmaking. The course was run by two teachers, one for a Photo class, who had been an actor, and knew about lighting, set-crew, and that sort of stuff, and another who taught the regular Video classes. I'd had both classes the year earlier, so I knew most of the stuff already. Halfway through the school year, the video teacher had accepted a job at the BoE, which he thought was starting in June......turns out, it started right then.
The photo teacher was trying his best to run it all with the new long-term sub, but he didn't know the technical video stuff, and the short classes on them earlier in the year hadn't been enough for everyone. Since he knew I was working for the TV station, and that I had had the video teacher's classes the year before, he asked me to help him out. Our big, end-of-the-year project was a news show-style video, encompassing events based around the school. We were each assigned/chose one-several different events to work on. One of these events, which the photo teacher filmed, was a FIRST competition, held at a Soverign Bank Arena in Trenton. Our team (Team Zero Gravity, if I recall.) had placed 8th, but it was still a good show. (The contest for that year was using the robot to lift a large ball over a divider bar in the middle of the arena the most times.) This was one of my individual projects.
Now, I don't know if you have acess to someone who can do this for you, but this is what I did. After editing down the footage, and removing most of the sound, (The audio recorded kicks in at the perfect time right when the crowd starts cheering for the competition starting, and hits a beat on the background music too; I love audio-synching, which is why the choppy narrating caused me much annoyance. It wasn't the reader's fault though, something happened between camera and conputer, it happened to everyone's footage from the handheld camera's, and not even the station manager could figure out what was wrong.) I had another classmate read some lines as narration, detailing what FIRST was, what it stood for, things about our individual team, etc., then layered them on top of the background music throughout the video. (Ending before the regular audio kicked in, to avoid a cacophany.) Initially, this video was just to be used for the ASIF class, but when we showed it to the elec./robot. teacher, who was head of the FIRST team at our school, he asked me for a bunch of copies of it. (According to the photo teacher, when the elec./robot. teacher watched it, he had tears in his eyes. Given the photo teacher's personality, I have yet to determine if this is true or not.)
I thought he was just going to hand him out to the class or something similar, but instead, he used the video as an explanation to possible sponsors, to get them more support. Having something tangible to show them like that really helped, and they've gotten more sponsors using this method. If you have the oppurtunity to make a video like this, you should go for it. Another piece I worked on, which was never put into any project, was one of the Seniors who was in FIRST giving a detailed speech about the FIRST program to possible sponsors, parents, and even some of the Elementary school and the like; demonstrating the robot, showing the history of FIRST, and that kind of thing. Even if you could just record something like that, it'd probably help out a bunch.
This was quite long-winded, when all I needed to say was "make a video". -_-; Anyway, I hope you do well in the Competition, and gathering sponsors. Good luck!