System.TypeLoadException: Method 'get_xxx' does not have an implementation

OK, so bizarrely adding a reference to App.Web in Api.Web and removing it again has solved the issue.

I have no idea why, but it did.

I changed the version of App.Web to 1.0.0.1 and the error was still showing 1.0.0.0, which is what prompted me to do it.

I wish there was a more reasonable explanation but there isn't. Such an infuriating issue i'm just glad to be done with it.

Best of luck to anyone else who experiences this, my thought's are with you

There can be many reasons for this, all the previous answers represent a case of this problem.

What I suggest doing is:
while your program is running open Resource Monitor -> CPU tab and in the search handles input box, search for the assembly that supposedly doesn't implement that method.

In the search results you'll see the path of your assembly, and most likely the path that you see isn't the one that you expect.
Delete the assembly from this unexpected path so that the correct assembly gets loaded.

For the records, in my case this was caused by two projects referencing different versions of the same package. At least fixing this solved the issue.