Sorry, I don't know of any contests and even if there were your chances of winning are probably statistically not good. Is the computer going to be yours to own, or is it going to be church property? It's not clear from your post.
If it's going to be church property then why not go to a local business or law firm and ask if they want to sponsor the program. My uni does this all the time. My department got a whole new computer lab this way a few years ago, and the business school got a new building.
Just tell them you'll put up a sign nearby saying "computer kindly donated by..." and they may jump at the chance to get some free publicity while also doing something good.
If on the other hand the computer is going to be yours to own and you just want to use it while doing the program, then why not put up a sign on your church noticeboard offering to do gardening, computer lessons, or whatever other service you can think of to earn some extra money.