We ran a contest a few years back and were surprised to learn about the complexity.
Online contests, done to the letter of the law, are a complete nightmare. There are state laws (yours and theirs) plus a litany of federal laws with the FTC, the Postal Service, and probably other agencies. And then there are privacy laws. And so on.
Our solution? Find a contest put on by people with real money, model yours off of theirs, lift their contest rules, and then run a fair contest. Just to be extra safe, we used UPS or FedEx for the prize.
The whole point of all these laws are to (1) protect the consumer from deceitful practices and (2) prevent lotteries/raffles. With a small, bona fide contest, who's going to complain to the authorities? And, since any complaint would be baseless on the surface, what agency will dedicated needed resources to your little contest?
As for taxes... the recipient is responsible to pay, and you should 1099 them.