My almost 4 year old is very awkward socially. She does fine around adults, probably because she is an only child and spends most of her time around adults. However, when it comes to other children, she just spends a lot of time staring at them. She has been going to a dayhome since she was 2, so she is getting socialized daily now. She is shy, and very smart, likely introverted. Should I be concerned?
if she is 4 she is in VPK right? also find local “story time” kids needs kids to.play with 
she needs a friend
No, you should not be concerned.
Nor should you pressure / make her socialize if she doesn’t want to.
Adults are allowed to be introverted / extroverted. The same goes for children.
No. There’s nothing wrong with being introverted