What vegetables continue to produce in heat above 100 degrees?

The only one I know of is okra, which I highly recommend for hot weather and clay soil.

If you know of any berries, or pretty much any edible that is shrub-sized or smaller (or is a vertical climber), that will produce in that kind of heat, let me know.

Have you tried sweet potatoes? I cook the leaves as greens...

What about cherry tomatoes? tomatillo, ground cherry?
Hot peppers?

Here's a link to some suggestions from your area Cooperative Extension:
https://travis-tx.tamu.edu/fil...
In case you're not familiar with Cooperative Extension, they have a lot of gardening information that is very specific to your local area.
Happy Gardening (doesn't have to be summer)!

Also... maybe try maxixie... sorghum... sugar cane... cow pea...
How about watermelon, or peanuts?

Maybe consider some other island crops like cassava (or yuca)...
https://www.epicurious.com/ing...