The Logelium page of Bernd Karl Rennhak claims that there are 637 convex polygons that can be formed. ThereThere are linked pages with pictures of the different types of polygons and the numbers of solutions.
edit: This is not a complete answer to your question. The ones depicted in the pages linked to above are built from the "simplified" tiles of the Stomach puzzle and furthermore satisfy the restriction that the tiles "conform to the base grid". I couldn't find a precise definition but this is described roughly at the bottom of this page.
It seems that Rennhak found these solutions via his software which is capable of finding solutions to tiling problems that fit on an integer grid. See these pages for some more details.
In any case it would probably be worth writing him if you are interested in more of the missing details.