You should be able to wire a relay onto the legs of those switches and control it that way. Take a screwdriver, bridge two legs and see if the button activates, if it does and esp32 and a relay board should sort you out nicely.
Hard to tell from the first photo, if you take the board out of the housing (looks like it’s just held in with plastic clips) and turn it over to see the actual buttons you can possibly wire in relays to press the buttons.
