A Sailrite walking foot sewing machine is what I use for vinyl, Sunbrella, and leather with a #18 needle typically. Otherwise I have a nice Brother industrial machine for more cloth projects. It has a BIG motor, but no walking foot. I got a lot of help from a close friend to redo my boats interior. Remaking the seat bases wider to accommodate more or larger butts from fresh wood, sealing them in epoxy, and making thicker foam cores. Do use marine foam that won't absorb water, and it cuts easily enough with a sharp bread knife. The thicker the vinyl, the less it will bunch up. Polyester thread won't degrade as much in the sun.
Staples are not the way to attach it. Best is to use a zipper and go all the way around the seat base. Starboard would be preferable over wood, but wood sealed with epoxy will outlast the original seat bases.
Sailrite has a ton of information on their webpage and they have a Youtube channel showing how to do about every sewing repair on a boat.
https://www.sailrite.com/
Look at the top for "Learn"