Before you leave the car, explain to her that she WILL sit in the cart seat, and she will NOT walk. Bring her a snack, sit her in the cart, and be firm. She's not big enough to walk (contrary to what another poster said, I cringe when I see 1yr olds walking in stores). My boys both rode in the cart until around age 3, when they could walk along at a good pace, could listen and pay attention, and then could be treated like 'big boys'.
Just keep at it and don't give in, because once she finds your breaking point of letting her walk, she will fuss every time until you let her walk (something I see happening to my friends all the time). Snack or treat in the cart, let her know before leaving the car that she WILL sit in the cart.
My boys have always shopped with me since I'm a stay at home mom, and because of it they both have incredible manners, lots of patience, and are very good boys. They know how to act in public, and i don't agree with always leaving kids home, they need to learn how to act in public, and always leaving a child home teaches them nothing. Bring your kids with you... its good for them, and you!