There is should be no problems breeding a 13 YO mare for the first time. I just bred a 14 YO maiden and she took the first time she was covered.
Star bred two 15 YO maiden mares and a 16YO without complications and all were successful.
Take the mare to the vet and have a breeding exam and clutures drawn, find the stallion you wish to breed.
You will probably want to have her USed to ensure she is pregnant then wait for the foals last trimester to begin feeding her extra.
If you choose to give the recommended shots at 3,5,7,9, months you can.
You can spend thousands getting the foal on the ground or you can do it cheaper by using live cover, and other less expensive things.
Remember it is the quality of the care not the cost that is important.
There is nothing written in stone about what is required .
There are plenty of threads here and most have great advice for the small breeder. Good luck. Shalom