How can I take care of my PC while it no longer has a tower wall?

A decent aftermarket case costs $40

If this is a prebuilt PC, then the case is cheap, and you would be doing yourself a favor by replacing it. You might as well replace it instead of trying to take it to a shop.

Some Cryptocurrency miners use desktop PC parts instead of using ASIC's, and in which case they zip-tie the parts on to Wire Storage Racks that you can buy at The Home Depot. These rigs last for years, no problem.

I have actually taken off the side panel on my PC on purpose to improve airflow until I was able to get a broken fan replaced. I had it like that for over a week without any issues. It doesn't harm your computer to have the case open like that. Just make sure that nothing can get inside to damage the circuitry. As long as you regularly blow out the dust with a can of air you should be fine. Keep liquids, food and pets away from the open tower. If you are still concerned you can always use ductape to reattach the broken side panel. Ductape is available in all kinds of colors and designs now, I'm sure you can find something to match your case or maybe even personalize it with a cool design.

Use pliers to bend the prongs/tabs back into place and then if the side panel will not fit back on in the normal way use gaffer tape to reattach it. Use the tape to seal the edges around the panel as well to make them dust-proof and air-tight.

Ugly, but cheap and functional until you choose whether to get a replacement casing.

While you can get by with that side panel for some time, it's HIGHLY RECOMMENDED to replace (or at the very least, repair) the case for the long run.

The biggest concerns will be cooling (as the missing panel will negatively affect the current system) & the potential of dust & other debris getting into your system components.

For cooling, try to keep the room a bit cooler... if possible (but above the dew point, as you DON'T want condensation) to compensate. You may take the "redneck" approach by slapping a box fan on the side to try & help with cooling. Just make sure you know which direction of the air flow in your system & if your system runs on a positive (pushing air out passively), negative (pulling air in passively) or balanced air pressure. The extra fan may not run as fast or as well as the built-in ones for your system, but it should help to some degree.

As for dust & debris, this is more of a monitoring situation. Just keep pets & other loose stuff away from the system (as computers tend to naturally draw that stuff in like a magnet).

I hope this helps in the short term.

Buy a new case.

A computer repair service can likely repair the damaged panel or order a replacement if not. Having it missing will expose all of the electrical components to dust and probably affect the fans from delivering cooling air like they are supposed to.

It would be really hard to bend the tabs on the wall so much that they couldn't be repaired. See if you can find a mechanically-oriented friend with a needle-nose pliers to fix it.

In the meantime, see if you can put that wall sort of in place, or tape a piece of cardboard or even just plastic sheet over the opening. Especially with small towers built by major manufacturers, cooling may depend on the cover being in place. The worst that will happen with a modern PC is it will do a thermal throttle to reduce heat and slow down, but that could be annoying.