I think this can be a great idea, especially if you set up some
parameters, which can be expanded as she gets older. We did something
like this with my ds for his 4th bday this year. We still surprised him
with a gift form us and a gift from baby, but we also gave him some money.

My mom was in town and there was a specific product she wanted and I
found a craft store which carried it, so we were taking a special trip
to this one particular store. I looked them up on the internet and saw
that they specialized in rubber stamps and stickers, two things ds is
crazy about right now. So in the car on the way to the store, we gave
him a little card with $10 so he could choose some items for himself for
his bday. While I helped my mom get her thing, ds went around with dh.
We showed him how to look all around at everything they had, then how to
narrow down his choices (such as deciding that with his amount of money
he could get one nice stamp and some stickers). They chose the stamp,
then counted down how much money would be left as they chose stickers.
He kept the bit of change to put in his piggy bank. This all went very
well, and ds was very pleased with his purchases and proud of himself.

So I would recommend choosing a category of thing to shop for and
choosing one or two stores to shop in, have this all decided beforehand,
otherwise random shopping in a huge mall with endless choices could be a
bit overwhelming. If you want to give her more choice of thing, maybe a
few days beforehand, say you'd like her to choose whether she wants a
doll or a game (whatever), then you figure out the best place to shop
for that item and keep her focused on the decision she already made and
how much she has to spend and help her find the things in that price
range to choose amongst.

-Karen, mom to Henry 4 and William 4 months-