I'm running out of space on my drive when I'm rendering the video and would like to know if there is a way to change the location of the temp/cache folder. Here's the error I'm getting.

2022-04-05 20:44:38 [DEB] Error writing trailer of /var/folders/zt/_73wsc_j0xv54qhw_0_zstdr0000gn/T/LRT_Render_5510850493225298341.mov: No space left on device
2022-04-05 20:44:38 [DEB] video:75616969kB audio:0kB subtitle:0kB other streams:0kB global headers:0kB muxing overhead: unknown
2022-04-05 20:44:38 [DEB] Conversion failed!
2022-04-05 20:44:38 [ERR] External encoder exited with error 1

I know using an external drive isn't the best option, but it's all I have for now. Is there a way to change the temp/cache folder to be an external drive?

The temp folder is in the location that your OS provides.
By default it's on the startup drive because that's often the fastest.
I'm not sure if you are able to change the location that MacOS uses as temp folder, but honestly, I wouldn't mess with that.
I'd rather recommend to cleaning up that drive to make some space. You can move other assets (videos, intermediary sequences, photos etc.) to an external data drive.
LRTimelapse also would be using the temp folder only temporarily, for example while rendering. After the render completes it moves the rendered file to the folder defined as "output path", which can well be on an external drive.