This needs a LOT of cleanup and commenting, but I'll summarize the main steps to get this working for anyone now.
The USGS national map viewer (link) is the tool I used to get the elevation data. Should look like the screenshot below.
Use the box/point tool to drag a box over the region of interest.
On the left side under the Data category, select Elevation Products (3DEP) and make sure ArcGrid is selected. I've been using the 1/3 arc second products, but I believe the other categories should work too. Some are only available in certain regions, but from my experience 1/3 arc second is available everywhere and it still has a resolution of 30 feet.
Click Find Products. It may take a minute to load the products in the region.
Once they're loaded, you can click Show Thumbnails to ensure that the data covers the entire region (shown below).
Then click save as Text to save the download links to a txt file. Rename this file map_zips.txt
map_rayshader_setup.R is the setup script you'll need to use. You just need to set the path to your directory and make a data folder. Details are given at the head of the file.
Within server.R you'll just need to adjust the path again to match your setup and you should be good to go!