I’m allergic to eggs. It’s not horrible, but if I have more than a cup of eggnog I itch all over for days. Am I allergic to the yolk, the white, or both? Can I make my own nog out of EggBeaters and be OK? I can’t stand the Silk kind, it’s think and tastes lousy, IMHO.
An ex was allergic to both. So, I assume that you could be allergic to either, or to both and that only tests will tell.
I would add that she became increasingly allergic to the stuff. Originally, she could eat, say, an omelet and only get a strong rash. Nowadays, even a small quantity will get her in a hospital for a week.
IANAAlergist. That said, if you want to try a homemade scratch test, separate and egg white and yolk, give yourself two small scratches with a sterilized sharp thing, and apply a bit to each scratch. See which one you react to.
Alternately, separate the yolk and white, cook them separately, and eat them a day or so apart. That should pretty easily tell you which is which, but as clairobscur mentioned, allergies can worsen with increasing exposure, so you might want to avoid any unnecesary egg-eating.
As I understand it, EggBeaters includes the whites of eggs. If your problem is the yolk, that should be okay.
Be glad it’s only gotten that bad thus far. By the time I was old enough to remember, if I ate an egg, it came back up promptly. When I was 18, I tried egg custard. It came back up just a few minutes later. 
And I know it’s the yolk. I can eat meringue (on pies, or in other desserts) without difficulty. I can have baked foods that have eggs as an ingredient without a problem; I’ve always been able to do that (lurve lemon meringue pie - not lemon cream). I can even have a little bit of French toast, so long as it’s made with equal amounts of milk and beaten eggs (yeah, it’s pretty soppy/sloppy, but it lets me have some egg).
If you want to be absolutely certain, see an allergist, but I think you could conduct your own test by eating an egg yolk (no white) and wait to see what happens. Then eat an egg white, and do the same. Your body will tell you which it is, or if it’s both. But you might be smarter to just leave eggs alone, if you’re not prepared to see an allergist and get a professional opinion (based on empirical testing).
I do realize that the idea of giving up eggs is a painful one. That custard tasted like some “food of the gods” to me …