Most plumbers start indoors: they ask which fixtures are affected, open a cleanout, and run a cable or jet toward the obstruction. If water rushes out of the cleanout as soon as it's opened, the blockage is downstream, and the plumber may recommend having the tank pumped before going further.
When a clog keeps coming back, a camera inspection of the pipe to the tank can show roots, a sag, or a collapsed section. The Board of Health's rules specifically address Orangeburg piping, an old fiber pipe that tends to deform, so ask whether your line is made of it.
Routine cleaning usually needs no permit, but replacing pipe does. Plumbing permits come from the Building Department at Town Hall, which is closed on Fridays, so plan around that if paperwork is needed.