"Client from contest has requested the full refund for his project."

What's to stop a company from running a contest, requesting a refund for the project, and then using the contest entries without telling anyone?

I entered a contest that ended up with almost 400 entries and then the client said they didn't like any of the designs, so no winner was chosen.

What's to stop them using one of the designs anyways?

Basically this would be a violation of national and international copyright laws. I'm not saying it couldn't happen, but if it does and they get  caught, and the odds are strong that they will  get caught(who  doesn't know how to  use a reverse image search?), they'll end up paying a lot more in legal fees and settlements than they  ever would have paid upfront in the first place. 

Theoretically it could happen, and I am sure that every  so often it does,  but there are various laws and agreements in place to  keep it from  being a real major issue.