For the nrmp and such get involved in your specialty society or state medical association. These are often the people who nominate those who sit on the bigger national organizations. The actual process will be variable from society to society and depending on if you are a student, resident, or attending.
For the state licensing board, this is extremely variable. It is often a political appointment by the governor. Even still, getting involved with your state medical association is probably your best way to get your foot in the door.