So gummy bears are typically made with gelatin. At high enough concentrations, it will not melt at room temperature, and is quite bouncy.
I would buy unflavored gelatin, and make a batch with four times the recommended amount and see how that works for you. That concentration will likely get a little soft, but will not melt at room temperature. If you still need it firmer, add more gelatin. If it's too firm, add less. Unfortunately, I can't know exactly what texture you want.
If you want it to be even more stretchy and rubbery, then look into making gummy candy. I've used a recipe from Sugar Geek Show that's made in a microwave, and it's quite simple. The issue I had with it was with air bubbles in the mixture, but I'm guessing this could be prevented by using gelatin sheets and a stove top method instead. I also reduced the amount of liquid for an even chewier texture. That said the ingredient proportions worked really well for me. To summarize:
- 50 g water
- 50 g granulated sugar
- 21 g gelatin
- 71g flavored liquid like juice (I reduced this to 20g safely)
- 85 g corn syrup (or honey)
- 1/4 tsp citric acid powder (optional)
- 1-2 drops candy flavoring (optional)
Combine the water, sugar, gelatin, and flavored liquid in a pint measuring cup, and give the gelatin 5 minutes or so to bloom. Then microwave for 30 seconds at a time and stir gently until completely melted, then add the corn syrup/honey and other ingredients if using. You should have about 1 cup of gummy candy goo. Then you can pour the hot mixture into your molds and let it set. Pry the gummy candies out of their molds and allow to dry at room temperature for a few days if desired, flipping occasionally. They will shrink a bit if you allow them to dry, but this will give them an even chewier texture.
Using plant-based gums or cooked starch could also provide you with a gel, but I think gelatin might be the easiest way to go.