Easiest is to put in at Bradford and paddle up to Burlingame - no shuttle required.

You could put below the Alton Dam on the Wood River and paddle down to Burlingame. Take out it a Bradford, so a shuttle is needed for this option.

Shuttle is also needed for the Carolina Campsite - put in at Lower Shannock Falls and take out at Jay Cronin/Richmond.

I am not aware that permits are needed for the campsites. There was a time when permits were needed for large groups at parking areas, but they haven't done that for years.