Are veterans permitted to live in the on post Apartments?

I can talk about availability of housing on a specific post, each base has it's own policy. Generally it depends on the number of units available. Priority is given to those active personnel assigned, then those assigned near the base, after that it is up to base commander.

If a base has open base housing, retirees and government workers can live in them. I've never heard that a non-retired veteran being able to. They must have a lot of vacancies in their base housing at Tinker. It will all depend on vacancies and post policies.

Ohio - appears Wright Patterson is the major installation there. Here's a link for housing: https://www.housing.af.mil/Home/Units/Wright-Patterson/

Pennsylvania - Google says the biggest military base there is Carlisle Barracks which is the Army War College. It says qualified military retirees or DoD civilians as well as active duty:
https://www.carlislebarrackshomes.com/

The other three bases are Tobyhanna Army Depot, Letterkenny Army Depot and New Cumberland Army Depot.

Indiana - appears to have Grissom Air Reserve Base, NSWC Crane Division Base and Camp Atterbury
https://www.cranehousing.com/
https://www.military.com/base-guide/grissom-air-reserve-base/contact/grissom-inn/7488