I would hesitate to push her into something. Honestly, that socialization thing is made into a huge deal when it really shouldn't be. Your little one is learning about others & interacting with them on a daily basis, unless you all are hermits. ;) She doesn't have to be in a room with 10 other kids, out of her comfort zone, without mommy to learn how to get along.
I think you probably just have a reserved little sweetie who needs a bit extra help to feel comfortable. That doesn't mean she's going to be dependent on you forever, unable to do anything without you. On the contrary, the more support you give her, the better she'll be. If she knows that when she runs off to play you are close by she'll feel better about it. She'll be better equipped to gain independence because it's not a scary thing. Make sense?
None of my girls have ever attended a MDO or pre-k or daycare. They are, however, normal & well adjusted & completely able to get along with other children. My three year old has no trouble leaving me to do gymnastics. My 8 year old is able to walk into a gymnasium of children she's never met for our homeschool co-op & tell me, "I'll introduce myself. Bye!"
Follow your heart & your mommy instincts & you'll all do great. And don't let that nasty "S" word worry you. :)