I have never heard of having a team sponsor, but our booster club does solicit meet sponsors for each of the meets that we host. However, we prefer to personally contact each of our potential supporters rather than send a letter. The letter seems too impersonal and most likely is going to end up in the round file. We make a list of all of the possible doners in the community and post it in the gym as a sign-up sheet. Club members then select those names which they wish to contact. We have had very good success with local manufacturers and retail stores, medical facilities and professionals such as lawyers, accountantss, doctors and dentists. We also make up a fee schedule based on the size of the meet. If its just a small compulsory meet then we keep the prices low but if we are hosting a large meet then the prices go up. As an example we start with a 1/4 page ad in the program for $100, and for more money we can go 1/2 page add, banner in the gym at the meet, a permanent plaque on the wall at the gym, etc. We always try to have one meet sponsor who gets the top notice, which might be a full page ad on page 1 of the meet program, a big banner at the meet, a prominent plaque in the gym and introduction at each session of the meet. These sponsorships are our largest source of fundraising money. Good luck.
If you have read my other posts you will know that I keep harping on the issue of funds distribution, but it is very important that you are aware of the law. If your booster club is 501-3c nonprofit then by IRS law you must share all funds equally among members of your booster club. It is illegal to apportion your income based on the amount of money brought in by any individual member. And, no, I don't work for the IRS. I'm just aware of a booster club (in another sport, not gymnastics) which was audited by the IRS and found themselves in very hot water becasue they were allocating illegally.