I don't know of any reason why this wouldn't be possible, but I'd like to make sure there aren't any potential pitfalls or gotchas here.
Can a .NET 4 application depend on a class library that's targeted at .NET 4 Client Profile? Are there any potential problems that could arise?
The client profile is a subset of the full .NET Framework. Thus everything in the client profile is included in the full framework, so whether it's a client or web application you're building, you shouldn't run into any problems.