I think CF13's answer is spot on. I just want to add a little bit of information to support/expand on it.
I assume by "choke", you mean a significant event, like death or aspiration pneumonia, not merely coughing or gagging. The latter are actually ways to clear the throat (kind of the opposite of choking.)
Without going into too much detail, there are reflexes which work very well to keep us alive. Coughing out and/or swallowing vomitus to keep the airway clear is one of those.
People who die from choking on vomit are those who are neurologically impaired in some way. For example, heroin use, heavy intoxication, unconsciousness (not the same as sleep!), a stroke, or prematurity (born before those reflexes are perfected) can dull that reflex, and that's why that kind of death can happen in those populations.
Babies who are neurologically intact will clear their airway, even of copious amounts of vomit, very well.
Cases of fatal choking are very rare except when related to a medical condition. The number of fatal choking deaths has not increased since back sleeping recommendations began. In most of the few reported cases of fatal choking, an infant was sleeping on his or her stomach.
So to answer your questions directly,
Can an infant choke on spit-up?
Yes, in the same way that it's possible for your spouse (let's say they have gastroesophageal reflux) to do so.
Do we need to worry about her choking so much?
No. If you're not worried about your spouse choking in their sleep, you needn't worry about your baby, though worrying about babies is very common, normal, and natural.
What if she spits up while we are sleeping?
Usually when babies spit up in their sleep, they simply swallow it or it ends up on the bedsheet/child's collar, etc. No one even wakes up. But if she's having a rough time of it for some reason, her cough/distress should wake you up. If you're sleeping in a different room (not recommended practice according to the American Academy of Pediatrics), please use a monitor set on 'loud' to detect distress of any kind. Room sharing done correctly is the recommendation.
Study: More Research is Needed About Room Sharing Effects On Infant Sleep, Sids Prevention