Don't worry. She'll be fine. My daughter was the same way when she was an infant; in fact, she wouldn't even let my husband hold her for about 2 months! It really sucked! She would go to me, our sitter (who has been here 2 afternoons a week since she was 7 weeks old, and one of my friends ;) I think that they had to be people who were very much a part of her routine (we went for a while there where my husband just never really saw her--then he started waking up with her in the mornings and since then she's a total daddy's girl).
I'm guessing that the first few days with grandma might be hard for all of you, but once grandma is part of her routine all will be well. I've always discribed my daughter as a "cat". She loved playing with people if they were sitting with her on the floor, but didn't want to be picked up/held by most people. It took my mil FOREVER to accept that. My daughter just turned 3 and is a wonderful person--very warm and loving, though it still has to be her idea to go over and love people. Isn't it amazing how much personality the have from the get go!
Good luck and remember that most things at this stage are phases that she'll grow out of. I hope that your family can respect her personality and love her for exactly who she is as I'm guessing that she's wonderful!