Mine does. It's very peculiar behavior but I guess it's something that baby's do. Licking handles. It's so odd. Funny thing is... I didn't have a problem with my first three kids gnawing on shopping carts. Maybe I carried or wore them more when I went shopping because it wasn't until recently, with baby #4, that I needed a shopping cart cover. Of course, it had to be modern, hip and not something that Grandma knitted or whipped up with leftover fabric scraps (she likes to do that).
I picked up, for a steal, the Numbers Cloud Cover (top photo above) by Infantino on Zulily a while back ($29.98 @ Amazon) and it's been great. It's puffy, cozy and covers everywhere Hadley could reach. Doubling as a high chair cover is also very cool. Yes my baby licks high chairs too. Good golly, doesn't she watch the news and reports on high chair germs? Yuck.
And of course, you can find more amazing, adorable and handmade shopping cart covers on Etsy.