Is it possible to pop a blood vessel in your eye while sleeping?

Maybe. Eye rubbing (especially when waking), pressure on the eye face down, or laying on your arm.
Maybe. We see a number of patients whose complaint is awakening with a subconjunctival hemorrage. This is most often caused by dry eye that can be exacerbated while asleep. Try using artificial tears prior to retiring - if they recur, consider getting your blood pressure checked and if indicated, check blood clotting tests.