Can I bake a regular cornbread recipe in a madeleine pan, or must it be specifically adapted?re? I

Another point of view - it depends on what you want.
If it's cornbread (traditional, more depending on baking chemicals for rise) in the shape of madeleines, any recipe will do....and follow HalfPint's suggestions on length of time to bake.
Or, if you want a pastry with cornmeal and more of the texture of madeleines (a genoise batter, dependent on beaten eggs for lift and rise), then use one of the cornbread madeleine recipes.

You can use any recipe. The only adjustment would be in the bake time. Likely to be much less than the time for regular muffins. I would use the time for regular madeleines as a starting point.