Bottom line up front follows:
If you are interested in starting the process, talk with an Army Medical Recruiter i.e. someone who specifically handles recruitment of doctors, nurses, dentists, etc in your area. You can do this before starting veterinary school or early in the first semester as the selection board for the HPSP scholarship is earlier than when I competed for it. They will help you with checking your eligibility for application (i.e. do you have a medical condition that would preclude serving) as well as collecting everything you need to submit a packet on-time.
Feel free to private message me if you have questions you don't want to necessarily share with the forum.