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sdb421

frogkisser

I don't have any specific product recommendations, but you might check Etsy.  There are a lot of fantastic work-at home moms who make products like this.

love2b_mrst

I have a floppy seat and love it. I got it on eBay for $20. 

blondii428

We registered for the Eddie Baur one at Target, looks nice and doesn't break the bank

cast920

The fisher price high chair cover is about 10 bucks on amazon. It fits best in wood high chairs but also works on the plastic ones. It folds into an attached pouch and doesn't take up much room.
For a cart cover, you might want to consider just making one. I made one out of fleece based on a pattern I found online. If you want it to fit the bigger carts just make the overall size a bit bigger. I used fleece so I wouldn't have to sew around the leg openings since fleece doesn't fray. If you fold over and sew the outside edge you can add some elastic so it stays on the cart better.

Unknown

My friends use the Itzy Ritzy covers, I am getting this one:

https://www.buybuybaby.com/store/product/itzy-ritzy-153-ritzy-sitzy-153-shopping-cart-high-chair-cover-fresh-bloom/1040062099?Keyword=itzy+ritzy