(disclosure: it's been ~10 years since I last had direct first-hand knowledge of this).
Giving away something for free in a "no purchase required to win" sort of promo is generally not a problem. The giveaway should generally be cheap (less than $600, or maybe it's $2000 now)... anyway, less than the amount that would be required to file a tax form in conjunction with the giveaway. Prize winnings are supposed to be considered taxable income, but at a low threshold point it is not an issue. In your case, I believe you're not going to be able to afford a big enough prize for this to matter :)
The contests where you have more issues are the lottery-ticket type systems where people pay to have a shot at winning a prize. The two big issues there (again, see disclosure above) are that you generally have to have some sort of non-profit status, and you have to have ownership/control of the physical prize before collecting money (you can't use a portion of the ticket sales to purchase the prize and then keep the difference).
For what you're describing, I think it's a good idea and easy to implement with no complex overhead or liability.