Think I'd save restore for a last restort ... other wise, however, Emma is correct ... one item to check is the ownerships/permissions on directories in moodle code directory. Updating plugins with version 3.2-> now wipe out those directories and then rebuilds them from a downloaded upgrade zip from moodle.org. If the directories permissions are locked down and the user under which your site runs cannot remove directories and contents + create new directories, etc. upgrades to plugins will fail.
See if you can't navigate to admin areas via URL line
After logging on, even if you experience the white screen of death, use the URL
line of your browser to see if you can't get to some admin areas that might relate:
Keep in the url line the path to your moodle .... http://site/dir/
add to the end of above the following:
Plugins overview (try this one first ... if you can get there you'll have
info to share back here:
Available updates to plugins
Main screen for 3.2 for Admin area
Check server environment:
The above might tell us that server doesn't have what it needs for
php or some mysql settings, etc..
As far as who to contact, depends ... where are you hosted? ... help desk tickets? Do you have a 'cPanel' or other interface into where your site is hosted?
'spirit of sharing', Ken