You probably won't be allowed to walk straight in. Look up trainers at your local track - you should find a statics table on equibase and contact them directly. If it is something you want to be involved in, I wouldn't leave it on the long finger. We're in March now, and our first turf meeting this year is the 23rd and it will kick into gear fairly sharp from there.
Explain to the trainer what you want to do, but the majority of successful trainers have worked and had an interest in racing from their early years. The very successful will have worked up the ranks, hot walker, groom, assistant trainer, and up to trainer. If you do go ahead and do things right - you want the foal to be trained properly. The here a serious difference between a proper trainer and a backyard trainer - and this will show on the racetrack. If you want go breed your mare, you want the foal to live up to it's full potential, otherwise you'll just have a TB that didn't make the cut and we all know there are plenty of those hanging around.
Stop for a minute, open your mind, learn. You may not agree with what I say, I may not agree with what you say but we will both learn something new.