Rules change with the JFK-BKK nonstop. You will arrive at BKK in late afternoon and will presumably have slept at least a few hours. Continuing on to Phuket should be an easy proposition.
The Chatuchak (I've also seen it spelled as Jatujak) Market is massive - thousands of vendors and scores of thousands of shoppers. It is also very hot, which can lead to lots of complaints from kids. (If they're fairly independent, I recommend handing them a couple of thousand baht, and sending them off on their own to see what bargains they can find, after establishing a time limit and meeting spot. It beats having to follow mom and dad around.) The Market can require a lot of stamina - I tend to outlast most of my Thai friends, but they usually go with specific objectives in mind and once they've acquired their items they are ready to go, whereas I like to wander around and take my time, making sure to take regular refreshment breaks. You can buy virtually anything at the Market, from lingerie to lemurs (the latter NOT recommended), furniture to ficus trees. And there are a number of places where you can ship the items back home without having to drag them back to your hotel. Generally the prices are quite good, so not as much haggling is expected or necessary, unlike any place with the name "night market" or "night bazaar" attached to it. Whether the Market would be worth your time is up to you, but I know many Thai and expats who specifically make weekend trips to BKK to stock up on household goods and clothing. (For those specifically intested in clothing, you might also want to check the markets in Bangkok's garment district, Pratanum.)