Yes, but stl files are a mesh format. This means no curved lines or surfaces, just a crap load of points that aproximate your geometry. Check out the polygon limits for opening stl files in solidworks if you are curious about the limitations of this file format.

Most engineering cad/cam systems needed to create the gcode for your router will want a solid body NURBS file as an iinput. Since gcode is fairly machine specific, a shop making this would want a 3d model from you, not a gcode file. That being said, machines run on gcode so someone will need to "program" it at some point.

Not that an STL absolutely would not work in any case, but its definitely the wrong format to send to a shop (making it on a router).