plastic fabric question

Hi, I have a high-chair I'd like to re-cover for my grandson to use here. It's a wooden one that converts to a table / chair & it's probably not worth doing as 2nd hand ones so cheap but I do have time & it's very nice.

I want to use waterproof fabric, I know Dunelm sells it on the roll. The old covering is suaiged together at the seams. Do I sew it? Glue it? Buy a heat gun & melt it together? I'm sure some of the clever crafters on here will know what's best to do....