Well it may work, and it may not. I'd buy seeds also to hedge your bets, for sure. The problem is that many newer varieties of vegetables are hybrids, and hybrids don't tend to reproduce very well. Their offspring tend to be weaker than the original plants. If these are standard grocery store veggies, they are almost certainly hybrid. If you have some old fashioned open pollinated veggies or heirloom tomatoes (farmers markets and CSA's are more likely to have these) your odds of getting a good harvest are better.
However, if it were my kid's idea, I'd probably do it anyhow, just for the involvement/enthusiasm of a child, kwim? I'd also grow some from seed, though.
I have had really good luck, however, with potatoes (just rotten potatoes from the store that sat in my cupboard too long then got buried in my garden) and pumpkins (ie I threw out last years pumpkin in my compost pile and a pumpkin patch emerged). So maybe try those.