As far as I know, there's no shuttles in any of those areas, but I can give you the numbers of the NJT routes that serve those areas.
Short Hills has the #873 to Chatham, and the #70 to Summit, running along JFK Parkway. It also has the #31 to South Orange, running along South Orange Avenue.
Summit has the #70 running along Broad Street to the Summit NJT station.
Chatham has the #873 running along Main Street (which runs parallel to the train tracks, so if you're not walking distance to the train, you're probably not walking distance to the bus)
New Providence/Murray Hill/Woodland Park have the #986 to the Summit NJT station.
Maplewood has its own shuttle covering the eastern portion of the village (in addition to NJT routes)
Maplewood Township, NJ - Official Website - Jitney Permits & Parking Permits
I also found a map of it:
https://www.google.com/maps/d/viewer...vteqCkvs&hl=en
West Orange has shuttle service:
https://nj-westorange.civicplus.com/index.aspx?NID=595
South Orange has shuttle service:
South Orange Village Parking Authority
Montclair has shuttle service:
Montclair Shuttle Service
Bloomfield has shuttle service:
Bloomfield Shuttle Service | Bloomfield Township
Glen Ridge has shuttle service:
Glen Ridge Borough - Jitney Bus Service
All of the areas that have shuttle service also have regular NJT service. Routes and schedules are available here:
New Jersey Transit