Is there an FCC label on the remote? Sometimes it’s inside the battery compartment. Alternatively, it might be an IR remote, in which case there won’t be an FCC label.
There are some people who have done various things with different models of adjustable beds in the past, you can search the forums for those, but it really comes down to the specific model.
The other option, which has the advantage of not voiding the warranty on the bed and maintaining any existing safety features is to use a automated button pusher, essentially a robot finger, to physically push the buttons.
I use these myself because I am quadriparetic, so I have one on a blender button, are used to have one on the microwave before I got a voice controlled microwave, I have one on the eject button on the DVD player.
They work fine, they just can get expensive. You will need one button pusher for each button that you want to be able to push, plus the mini hub that gives you Wi-Fi connection and a SmartThings connection.
The Switchbot brand has an official smartthings integration which works quite well. So you don’t need any custom code. But the button pushers are typically about $35 each (sometimes they’re on sale for $30), and the mini hub is usually around 40. So, like I said, not cheap, but pretty straightforward.
Here’s the review I wrote of these for this forum:
2020 Switchbot Review ( robot button pusher), integration through IFTTT or new V3 app
So if you find out that you can integrate directly via RF, great, but if not the button pusher is another option. 