I haven't eaten all these, but I'm trying / want to try:
Malabar gourd, Cucurbita ficifolia - immature fruit as summer squash
Wi apple, Spondias dulcis - immature fruit as pickles
Tonkin jasmine, Telosma cordata - Wikipedia says "The top, fruit and flowers can all be consumed as vegetables."
If we're counting young bean pods, we can add lablab and perhaps lima and Clitoria ternatea and Bauhinia purpurea and Leucaena leucocephala and Parkia speciosa to Galatians522's winged bean and pigeon pea.
Katuk, Sauropus androgynus
Joint fir, Gnetum gnemon
West African okra, Abelmoschus callei is supposed to be perennial, as is the "Nkruma tenten" variety of A. esculentus. I think bele/edible hibiscus, Abelmoschus manihot, can also make okra-like pods.
Whatever perennial mallow species can grow in the tropics should make the edible cheese wheels. We had one Malva parviflora (I believe) here.
Immature fruits of some other Artocarpus species can be eaten similar to green jackfruit, I think.
Moringa pods
Many people use starfruit as a tomato-esque veggie. (And perhaps bilimbi?)
Akee, Blighia sapida may qualify as a vegetable?
Eggplant grafted onto turkeyberry, Solanum torvum, is pretty reliably perennial.
Gboma eggplant, Solanum macrocarpon
Fun exercise!