If you don't have enough money for the vet, you will lose 10 karma points. I don't think anything will happen to the foal, but I'm not sure (I always make sure I have enough money).
Tip: If you do not have enough money and you know your mare will give birth, don't look at the mare's page until you have enough. Time will not pass for your mare.
Get money from your job, stud coverings, and competitions.