Can we move a module from the directory "/modules" to the directory "/modules/contrib" without uninstalling it?
As with most things, I suggested testing it out on a development/test site before trying on the live site. It's likely that the specific set of contribs and the libraries they use will affect the success of this operation.

How did you install the modules? Do you use a composer workflow?

In general, some paths stored in different places are going to be wrong. To fix that, you will need to edit those paths or have Drupal figure out the new paths. I believe doing a cache rebuild (e.g. drush cr) will fix that problem.