I read most of the responses to this. Funny how everyone has an opinon on this.
Yes, it's tacky to ask for money and most moms would probably be turned off from going to the party as T.F said. But at the same time, unless I was anticipating your daughters bday and had a gift in mind, I would maybe be relieved to not have to go shopping, just pull a $10 bill out of my wallet.
I don't know what your bday theme is, but if it's swimming, then perhaps suggest in the invitation like Sandy S suggested that people bring swim themed gifts (pool toys, beach towels, goggles, etc). Or sometimes, if you write an invite from the childs point of view, people think it's funny, like "I'm having a bday party . . . Mom says that it's not nice to ask people for money for a gift, but I really want to go to music lessons this summer...."
If you decide not to ask in any way for money/gift cards/etc, then you could either try to sell back un-opened gifts, re-gift them to someone else later, or let her play with one new toy a month. My 2 yr old loves pulling a new gift out of the closet. Pulled out a Xmas gift just this week. He'd never played with it. What a treat for him!
Grandparents want to spoil their grandkids. So I agree with most of the other posters when I say to ask them or ask your parents fora cash gift, as long as they know what it's going to.