How to get Exception Error Code in C#

You can use this to check the exception and the inner exception for a Win32Exception derived exception.

catch (Exception e) {  
    var w32ex = e as Win32Exception;
    if(w32ex == null) {
        w32ex = e.InnerException as Win32Exception;
    }    
    if(w32ex != null) {
        int code =  w32ex.ErrorCode;
        // do stuff
    }    
    // do other stuff   
}

Starting with C# 6, when can be used in a catch statement to specify a condition that must be true for the handler for a specific exception to execute.

catch (Win32Exception ex) when (ex.InnerException is Win32Exception) {
    var w32ex = (Win32Exception)ex.InnerException;
    var code =  w32ex.ErrorCode;
}

As in the comments, you really need to see what exception is actually being thrown to understand what you can do, and in which case a specific catch is preferred over just catching Exception. Something like:

  catch (BlahBlahException ex) {  
      // do stuff   
  }

Also System.Exception has a HRESULT

 catch (Exception ex) {  
     var code = ex.HResult;
 }

However, it's only available from .NET 4.5 upwards.

You should look at the members of the thrown exception, particularly .Message and .InnerException.

I would also see whether or not the documentation for InvokeMethod tells you whether it throws some more specialized Exception class than Exception - such as the Win32Exception suggested by @Preet. Catching and just looking at the Exception base class may not be particularly useful.

Building on Preet Sangha's solution, the following should safely cover the scenario where you're working with a large solution with the potential for several Inner Exceptions.

 try
 {
     object result = processClass.InvokeMethod("Create", methodArgs);
 }
 catch (Exception e)
 {
     // Here I was hoping to get an error code.
     if (ExceptionContainsErrorCode(e, 10004))
     {
         // Execute desired actions
     }
 }

...

private bool ExceptionContainsErrorCode(Exception e, int ErrorCode)
{
    Win32Exception winEx = e as Win32Exception;
    if (winEx != null && ErrorCode == winEx.ErrorCode) 
        return true;

    if (e.InnerException != null) 
        return ExceptionContainsErrorCode(e.InnerException, ErrorCode);

    return false;
}

This code has been unit tested.

I won't harp too much on the need for coming to appreciate and implement good practice when it comes to Exception Handling by managing each expected Exception Type within their own blocks.