Might not be so easy -- it is like paying payroll -- you, not only the recipient, need to report the tax ID of the person/entity that you paid.
Right, so you provide that information to the winner to put on their form. No big deal.
Nope, you need to put the payee info on the forms that you submit as well as the recipient.
If you are paying someone more than $600, you need to cut a 1099 form, which tells the category, such as interst, rent, misc, etc. (1099-Int, 1099-Misc, etc.).
On the 1099, you must put the Tax Id of the recipient, which for a person is their SSN, and for another entity is typically their EIN.
You submit your copy of the 1099, and they submit theirs. They need to match or the IRS see a red flag.
So, no, you can't simply say "here's the form, you submit it".
Source: I'm not a Tax Attny, but I live with one, and have co-owned several businesses that needed to cut and/or receive W-2s, 1099,s etc.
Make the winner fill out the tax forms and send them directly to the IRS. The IRS informs the promotion they've received the tax forms, and verifies names, addresses, amount match. The promotion releases the prize to the winner.