Sorry for the late answer.
While it is possible (in principle) to enable users to install R packages within the running docker container, I am not aware of an easy way to enable these users to save the container as a docker iamge, which they could restart at a later time.
However, the question of reproducible environments is important to us. So important, that we have a separate sub-site for it: https://environments.rstudio.com, which lists several workable solutions. For example, you could use the snapshot, which uses the renv package to record the state of a project specific library.