Hi,
The simplest means of control is to use something like a Sonoff running a replacement FOSS firmware like Tasmota. There’s many hardware options, but to start out, a Sonoff 4CH R2 Pro offers 4x relay outputs with different models designed to power mains or low-voltage devices.
The simplest would be to use the Pro with volt-free relay switches to simulate pressing buttons on the remote - but you mention the remote is broken. If the remote were IR rather than RF, Tasmota supports sending IR remote control codes directly, but that’s not a good option.
Most consumer devices are built these days from Lego-like modules, and the photo shows a vertical daughter board with RF components and the aerial. With the remote, this will be a standard part with several low voltage outputs. You could even try buying random RF remotes from eBay and see of they might work with the receiver module.
The first thing to say is BE SAFE - this looks to be a high current mains device with a large 24V transformer. If you are new to electronics, reverse engineering a mains device may be hazardous - turn it off, wait for capacitors to discharge, and also be careful of moving motors removing appendages!
As the remote is dead it’s going to be a little harder to trace the outputs from the vertical RF received module to the 4x control relays. Board markings or circuit layout might help. The idea would be to remove the old RF receiver and replace it with 4CH relays. You could probably replace it directly with a custom ESP8266 like a Wemos Mini D1, but that needs a lot more electronics knowledge.
I’d expect the 4x relays to be in pairs to control two motors, and the relay coils driven by transistors or a driver chip (like a ULN2001). The back of the board might show 4x relay coil traces going to a driver, then the RF receiver module - remove the module, look for logic power and ground (I can see a voltage regulator on the PCB, so guess at 5V, not 24V), and if safe (check with a multimeter), connect one of the removed RF receiver signal pins to logic power (5V?) and ground - did anything move?
EDIT: After a closer look at the photo, R4, R5, R6, R7 seem to connect from the RF receiver module to 4x driver transistors next to the relays - guess where I’d start! 
Another approach would be to reverse engineer the motor connections to the chair and rig up 4CH relays to control these. Each motor will need 2 relays (basically forward and reverse), but you also need to consider any limit switches in the chair to protect you and the electronics.
Hopefully that’s enough to help you get started…
James
