I would do this by adding a universal garage door receiver to simulate pressing each button in your existing hardwired controller. The big question is if the existing buttons are low (12~24V) or mains (120~240V) voltage, and how you can hook into them.
You basically need a "universal garage door receiver" that's designed to interface modern wireless controls to an old garage door opener. These units are available from various manufacturers (be sure to check compatibility with your in-car buttons which are likely HomeLink). You'll need either two or three receivers (depending on if you want to press the "stop" button).
If your existing hardwired button controller is low voltage, then open it, and wire the control terminals of each receiver in parallel with each push button. This might be just a matter of hooking a wire onto a terminal, or it might require soldering a connection to a circuit board. You'd have to post a picture with internal details.
For any run within a few dozen feet, you could use 22AWG wire (eg, phone or security system control wire).
If your button control is mains voltage, it complicates the project quite a bit -- unless you can find a universal receiver with a relay output that specifically can handle mains (I couldn't).
You'd had to use relays to have a low-voltage signal for the wireless receivers, then let the relays control the mains power. I'd personally probably convert the existing button control to also be low voltage (control the relays) while I was at it.
I can post details about how to do this, but really the most 'difficult' part is just mounting everything in a safe way (proper-rated enclosures, relays, and wiring a panel safely), and since it's complex to explain all that and I don't even know if your control is even mains voltage, I won't type all that out now.