You can watch film of the 1929 fifth round FA tie between West Ham v Bournemouth at Dean Court (attendance 11,346). The game resulted in a 1-1 draw. Tommy Yews got the goal for the Hammers. He also got one of the goals in the 3-1 victory at Upton Park (attendance 30,217).

http://www.britishpathe.com/record.php?id=48808

Unfortunately, West Ham was knocked out in the next round by Portsmouth. Jim Barrett got both goals in the 3-2 defeat.

I wonder who scored the first West Ham goal actually filmed in its entirety ?

We see the ball hit the back of the net against Bournemouth but it doesn't seem to match up with the previous footage.

My guess would be the winning goal in the 1940 War Cup Final by Sam Small ..?