D.,
Hang in there! Your daughter will be just fine :).
My second daughter gags herself a LOT, and though its been difficult to deal with, she is doing just fine. She was born very premature and came home after 4.5 mo in the Nicu using a pacifier or my breast when she needed to be calmed down. Then we went to a baby shower for another new mamma and the new mammas mother, a very wonderful African woman, introduced my little Layla to the concept of sucking on her fingers. Now this woman meant no harm, where she is from they don't have pacifiers so that's what they teach the children to do. I didn't think much of it... it was only about a 2 hour baby shower and that was the only exposure to sucking her fingers Layla had had up to that point. Well when I got home I tried the pacifier and she refused it... and from that day till now ALL she wants is her fingers. It really has made breastfeeding, bottle feeding and spoon feeding a very big challenge.
Compounding this issue is that my daughter has problems with digesting her food... so when she gags herself on her fingers she projectile vomits then will sometimes choke on her vomit. There are many times she has turned herself blue as well. There are many times I've had to help her get her fingers or the vomit out of her airway. She has been doing this since July and even insists on sliding her fingers into her mouth while I'm trying to feed her (takes a bite but won't swallow till those fingers are in her mouth). My 2.5 year old has been incredible ... when we feed Layla she holds her hands down so all the food gets in her mouth and she can't gag herself. My 2.5 year old is amazing at entertaining her sister's hands while I feed her, I don't know what I'd do with out her help!
Since my older child did not suck her fingers it has been a very new experience for me, and while I was laid back with my first daughter, with Layla I've been a bundle of nerves due to her prematurity and many of her medical and developmental problems. With her fingers specifically she has gotten so much better at not gagging herself. Eventually they do learn. As I think about it my first daughter went through a short gagging herself stage but mostly when she was playing she'd scream for joy then reach those little fingers in to try to figure out where the noise was coming from. She learned so much more quickly how to remove the fingers in the event of self gagging.
With both of my girls I've either worn them in a sling/mei tai or had "baby stations" in each room of my home. In the kitchen it is a high chair (as soon as they are big enough to fit into it) or a bouncy seat when they are smaller. In the livingroom it could be the bouncy seat or the exersaucer. I also keep a moveable "station" available for if I need a shower but DH isn't home or for things like when I do laundry. This way my baby rotates rooms like I do and gets to interact with me ~ it helps her development and it allows me to be there if she gags herself... so I am able to get my house work done with out being worried or anxious. It is also more fun for me to have company while I am trying to do my housework :). I do let her nap alone and though she sleeps in my room now, she has no trouble while sleeping.
Your daughter will learn to take her fingers out of her mouth when she needs air. She will be OK :). Babies are incredibly resilient, so even if she forces herself blue or in a more extreme case she passes out her body will recover quite quickly.
I know it can be very bothersome, esp. if your daughter is turning blue at times, but it will get better!