I assume that you have a 64 bit ASP.NET worker. In this case, make sure that your ASP.NET project adds a reference to bin64\Esri.FileGDBAPI.dll and that Copy Local is set to true. Also, you need to manually copy the other DLLs in the bin64 directory such that they are in this directory.
My guess is that your windows form application is x86, i.e. 32 bit.