My first assignment was to loot indiscriminately a certain sum of money from...Raven Rock. Oh well, gotta prove my loyalty, I suppose. I paid my passage to Solstheim and started looting everything in sight. I know for a fact I've taken in value well over 1,000 drakes but the quest isn't updating to return to Delvin. Is this glitched? Am I going to have to report myself as a failure and resign?

Yes, this is a bug. Unfortunately, there is no way to fix it. The only options are to tell to Delvin that you want to quit the job or reload a save prior to accepting the quest.