Taking over an existing Drupal site
To save yourself access headaches:


Get credentials for the "user 1" account that has full Drupal access
Get database access or a copy of the most recent database
Get SSH Access to the production server


To audit the site once you get access:


Install http://drupal.org/project/hacked to make sure there haven't been any undocumented changes to Drupal core or contrib modules.
Look in /sites/all/modules/custom to find any custom modules. If nothing is there one good way to tell if you're missing any hidden ones is by looking at http://example.com/admin/build/modules and looking for modules without a version number. They may be hiding in /modules or /sites/*site_name*/modules.
Have a look at the theme and especially the contents of template.php. Often times a site will have a lot of modifications happening here.
After looking at the items found in #1, #2 and #3 you shouldn't expect to have found a whole lot. The custom modules should be mostly hook implementations and 'glue code' not re-inventing the wheel or re-creating functionality found in popular contrib modules but your mileage will vary here depending on how custom the site requirements originally were.