You don't literally "build" the plushes from Build-A-Bear from scratch. There's a pre-made outside shell that gets stuffed and you can customize the shell with accessories, or put a noise box in, and stuff like that. So, no, you can't build a Pidgey, or any other Pokémon; it's possible that they may add more Pokémon to the line at some point, but if they do, it would be only be more super-popular Pokémon like Pikachu is.
In the meantime, you can buy legit regular plushies of many other Pokémon at the Pokémon Center or via eBay or Sunyshore or other resellers if you need to get your "not a Pikachu" fix.