Wheel back chair is also a term used for the above chairs.

The current seat upholstery is not doing your chairs any favors. You may like them better with something snazzy on the seat.

Vinyl? Because of your children? If you like vinyl, go for it, but not because you expect your children to eat like Fido at the table. I can almost promise you that children emulate their parents. They will eat properly with minimal effort on your part. I have four boys and while I have had a few broken dishes and the occasional spill, those are the exception, not the rule.

You might be able to find a pretty patterned oil-cloth fabric which can be wiped down.

Here is a site that features it. Much nicer than vinyl. https://www.oilclothbytheyard.com/collections/fruit