Most simply, you can have a community drive. Make up flyers, and talk to local businesses, especially those that are owned by local owners (they won't have to clear things through their corporate office that way). Have them be collection points for donations and postage stamps. Pass out flyers in neighborhoods, at schools and churches (places like that you need to talk to a person in charge and have them pass out the flyers--make the school or church a collection point for donations). Also, see if you can contact your local newspaper and see if they will do a story on you and your project. Even the smallest mention is sure to get you lots of publicity and a lot of donations.
I should tell you, however, that unlike previous wars, the US government is not taking mail for general delivery this time around. During the Gulf War, I wrote countless letters and cards, and sent them for general delivery, and they were simply given to soldiers. I even got a few letters back :) However, I read recently that there is so much mail going to the Middle East right now, that the government has had to shut down that particular program. You must have the name and specific address for a person in uniform in order to send mail. All other mail is being returned.
The article I read about it, which was in my local newspaper, said that despite the ban on general delivery mail (meaning to just any soldier), the government is allowing care packages containing personal items like toiletries, snacks, candy and gum. For obvious reasons, things like chocolate are a bad idea, but you could still do your project and send care packages instead of cards. Obviously, your money won't go as far, but you can still do a drive for donations and postage costs.
Make up a list of the items you want to send, and be specific. Don't just say toiletries, say soap, disposable razors, deodorant, tampons, etc. Then, when you have collected your items, print off some little cards to go with them, saying they are from you and the people of your community. Make up little kits in gallon sized zipper bags, and mail them off. That kind of stuff does go through, and I am sure it would be greatly appreciated.
I didn't mean to rain on your parade, as I think you have a great idea. You just cannot send regular mail. But your idea of sending something for the holidays is really splendid. Just get yourself organized, talk to the local post office so you know what you can send and the costs involved, and then get started!
Hope it goes well!