A lot of Steam games have their own controller mapping settings, and while most will work without configuration, sometimes you need to go into the game options and change the controller mapping.
NOTE: We have had very bad experiences remapping things through Steam's interface, so we generally recommend sticking with the settings in-game whenever possible (note that some Steam games don't allow you to map the controls, so this will not always be an option).
For Pinball Arcade, you can go into the options / settings within the game and remap which buttons operate the flippers.