It's the worst -- that clingy, raised-arm, shrill-screaming, embarrassing mess some toddlers become whenever someone other than their parent so much as looks in their direction. Don't worry -- she'll get over it. Lila used to do this to everybody, including Nick's mom. Whenever she came around, my child buried her face in my neck and swatted away kisses. I'd dole out apologies, but luckily, Grandma didn't take it personally. She gave her grandbaby a little time to get to know her and recognized what I was too mortified to see: Toddlers, like babies, can have stranger anxiety. Some are more dramatic than others, but most eventually grow out of it.
Until then, don't try to force your daughter to interact with people she doesn't know well, like the lady at the grocery or a distant relative. Let your close family, friends, and neighbors know that your child's a little shy and you're working on making her more comfortable around them. Then take her out to places where there are lots of people -- the park or the mall -- so she can see you interact with others. Lila eventually grew comfortable with her grandmother when we moved closer to her; instead of seeing her on holidays, she saw her almost daily. Soon enough, Lila blossomed into a social butterfly, and Grandma got all the loving she could stand. Eventually, you'll see a change in your child, too.