Greetings! I have been allergic to dogs, cats and horses (among other things) since I was very young. Being a rebellious child, I told the doctor where to stick it (at the tender age of five) and proceeded to bring home all three in quick succession to my parents' dismay. I won't lie and say it wasn't heck on my allergies, but I was willing (and still am) to deal with them in order to have what I'd always wanted - a pet. Now, I will say that as I have aged (don't ask and I won't tell), my allergies have moderated somewhat but I remain allergic to animals. I completely understand your situation and empathize.
To some degree, your decision may or may not depend on how willing you are to put up with allergies to be with animals. For me, there was no question. I didn't care what doctor said what (my dad happens to be one of those doctors) - I was going to have animals and that was that. That would eventually translate itself into a career in veterinary medicine. While it may not always be so pleasant for those around me to have to listen to my snuffling, etc., for me it is the right decision and one I'd make again, in spite of what some critics might say. Pls. note that I do not advocate putting your life on the line though - if your allergies are too life threatening, then by all means, do not sacrifice your life.
Don't write off vet school yet. Like the person before me said - there are lots of vets out there who are allergic. 🙂 Good luck!