EIN's - the federal gov't in the health care legislation, was going to require a business (or even person) who pays someone more than $600 per year to report it on a 1099 at year end. This was (thankfully) repealed, but as a result, business are being more vigilant about collecting this information because they are cracking down more on businesses who aren't properly reporting payments currently.
"Donate/501c3/Write off" - What the sponsor is essentially doing is giving you money to market their products and/or promote them in some way. They cannot write it off as a charitable donation unless you are actually a charity, which you are probably not. Your sponsor can write it off as a marketing or general business expense. Think about NASCAR, DuPont gives money to sponsor a race car team, which is not a charity.
So when your sponsor says "write off a donation" they really mean they are giving your BBQ team money which you have to report as income.
You can deduct your reasonable BBQ team costs to offset this income, up to the extent of the income from your BBQ "hobby." You cannot use a loss from your BBQ "hobby" to offset income from other sources - like your salary from your main job, or investment income. If you want to read more about this search "Hobby Loss" or "Hobby Income" on the irs.gov site.
BBQ hobby income would also include your winnings from competitions.
If you are a qualified charity, then they must get a letter from you acknowledging the donation to be able to take the deduction. Otherwise they are just asking for your TIN/EIN to be able to issue you a 1099-MISC at year end.
As a side note - if you are running a catering business and a BBQ team, you have a for profit business, and you can write off the costs of the BBQ team as a way to promote your catering business, as a marketing expense.
Please check with your tax professional to understand your record keeping / reporting requirements.
Your Tax Payer Identification number (TIN) can be your SSN, because all income/expenses to offset income will flow to your personal income tax return unless your are a corp. You don't have to get a EIN (employer identification number) unless you are a LLC or you have incorporated or have employees.