The root path to the temporary folder location is determined by your OS, I'm guessing this is Windows in your case ?
https://www.askvg.com/where-does-windows-store-temporary-files-and-how-to-change-temp-folder-location/ see the second temp folder type, the one which is per user.
R provides a tempdir() function that makes a random available sub directory in thatyemp directory.
I wonder whether switching to using renv package would help. If you have a shared drive that could serve as a common cache for installed packages you could perhaps set things up where each user when using renv::install would first check for the desired package in the renv cache rather than the CRAN sites.