Create a start → stop/time on delay → reverse motion → stop circuit

You can program all of the functionality into the PLC. A simplified version could be:
1) Each target has a single relay to control the direction of the motor. This relay is controlled directly by an ouput of the PLC.
2) Two limit switches (capable of carrying the motor current) connect in-line with the motor and inhibit it when it hits a limit.

Note that there is no feedback relating to when the target hits a limit. You can compensate for this in the PLC software by adding the travel time to the dwell time and activating the PLC output for this total time (e.g. 3 sec. travel time and 9 sec. dwell - PLC output activated for 12 seconds). This configuration is obviously much simpler for you to connect up, is more robust and likely to work first go, and is configurable without you having to run out to the target zone and changing the DIP switches.

You can always connect the limit switches to inputs on the PLC if you do think you need that information.

An alternative to the PLC is a PC/laptop with I/O card/interface.

If you refine what you want the targets to do a little you could use an off the shelf controller like this;
https://www.frightprops.com/control...ntrollers/picoboo-controllers/picoboo-jr.html

There are many controllers like this used in the haunt industry.
No programing needed, just press a channel button for as long as you want it on/off