I can think of a couple off the top of my head....

Sugar snap peas/snow peas

Eggplant (variety I want to grow is bambino, a miniature variety; would I still need to train the vines via a trellis)

Melons (cantaloupe, watermelon; both varieties I grow would be bush) (???)

Do bush/compact varieties of tomatoes (known as determinate)
need to have their vines trained via the use of a trellis?

I very deliberately avoid using any kind of trellis whenever possible--untangling dead vines from a permanent trellis each fall is murder.

Instead I use cotton string and train vines to grow up that. At the end of the season, I simply cut the strings and throw the entire mass into the compost--strings and all. It's an easy five minute job rather than a couple hours having my hands rasped by plant bristles.

The plants that I train up the strings:

Pole peas, pole beans, cucumbers, bitter melon, vining varieties of squash, vining melons.

You should not need support for the eggplant--I've never met a vining eggplant. Determinate tomatoes should be non-vining, though I've had to stake a few in the past to keep them from flopping over. A few of my pepper plants as well needed staking. I suppose the eggplant could flop if the stem isn't sturdy enough, but it should not require trellising.