I searched by the chair make/model information and the circuit board model. No joy. So, here are my thoughts...
Firstly, you have a low number of posts... That leads me to think that maybe you are not an electronics hobbyist and that you probably do not have a lot of skills for working on the circuit. This idea could be completely wrong, of course. But, it would help to have some idea of your skills and how much time/effort your are willing to put into the project.
Second, I have a philosophy... If something is broken and doesn't work, me fooling around with it can not make it not work more... Meaning, it can't hurt to mess around with it. If you agree with that idea, then read on...
Thirdly, these chairs sometimes have a separate power supply box... Let's make sure that your chair has a power cable that plugs into the wall. Yes? If not, you have more problems.
If so, then the circuit board is likely to only do a few functions... Power regulation, chip control for different massage programs and switching the motors on and off/controlling their speed... If you can live without the different massage programs, tapping directly into the motors and switching them on and off would not be that difficult, assuming you have basic skills and tools.
I think I see a couple of relays. These probably provide power for the motors. Find some way to energize them and things are likely to start vibrating.
If you lack skills and tools, look for a "MakerSpace" club in your area. These clubs are made up of individuals that tinker with stuff. Electronics, computing, carpentry, metal works, mechanics, everything "hackable." Lots of them have amazing tools. All of them have helpful, friendly members. Additional resources might include the shop teacher at your local high school, or your local vacuum repair person (though they are going to want to be paid.)
The board and its functions could be completely recreated, of course, using a different controller chip and writing custom massage programs to suit your desires. This might be accomplished with an "Arduino" controller. Arduinos are controllers set up for hobbyist to easily control electrical circuits, by writing their own control code. You'd need a 13 year old, from the MakerSpace... lol.
Let us know your skills, tools and the amount of work you are willing to put in it and we will help you along. (The amount of work may be expanded, if you want to work on it for the sake of learning.) We are glad you have joined our community; welcome!