I need a PLC Programmer

I have a need to fill bags with pieces of wood. We have a conveyor belt with 3 pneumatic pushers spaced evenly along its length. Each pusher has an electronic eye to sense when the pieces of wood going by on the belt need pushed into the bag. Ideally, we would have a PLC programmed to trigger the pneumatic pusher the correct time after the wood has passed by the eye to get pushed into the bag. The pusher would push wood into the bag until the bag is full (X number of pushes to be experimentally determined and changed as needed), then that pusher would be turned off and wood would pass by and the next pusher would fill the second bag. When bag #2 is full on the second pusher, that pusher would turn off and wood would proceed to the third pusher for the third bag to be filled. Once the third bag is full, the first pusher would turn back on, and the cycle would continue. Attached is a highly simplified diagram of what the conveyor looks like.
I do not have the PLC's at this time, and would like support in selecting and programming the PLC's to fulfill this task.