My 3mo enjoys putting her fist in her mouth and chewing on her thumb. Is this part of her motor skills development, and should I pull her hand out from the mouth whenever I see her doing that? Sometimes, she will suck her fingers till she vomit as well. ?

Yes this is very normal and almost babies will attempt to suck their fingers or some other things. They are using their mouth to explore and secondly could also be baby is teething soon and need to soothe the gums. I will try to pull her hands out when she tries to suck and give her a teether. The downside of doing this is she will suck other things around her even when she is outside. But if you let her get into the habit of sucking fingers some babies will get hooked on it so much they will keep sucking fingers even when they are much older as it is harder to quit fingers. My niece suck hers till 6 yo! But if you are afraid she will suck other things which may not be as clean then fingers is better. Depends on your thinking. But not all babies are the same. For teether I love the Ange brand banana teether made in Korea :)