I wouldn’t use a paper stapler. The staples would get pulled out too easily. Plus, all the electric staplers I’ve seen staple when you put your stack of paper between the jaws. I doubt your chair seat will fit…
The projects I’ve DIY’d, I used staple guns that I got at hardware stores. It’s a bit intimidating to decide what staples you want to use (different widths, different lengths of the pointy arms, different degrees of heavy-duty-ness, etc). But it depends on thickness of the material that is being used, chair seat wood type and how thick a wad of the fabric might be as you gather it to go around a curve. Some wood is just soft enough that the staple bangs in like butter. Others may behave like you are trying to staple to a rock (crumpled staple, no joy) and sometimes, the wood just starts to splinter. But you’ll probably know if is that bad before you get to stapling.
I dunno whether hot knives are lurking, but assuming you’re not using foam the thickness of a mattress, you can probably just use a pair of utility scissors. (I.e. not scissors that cost $45/pair, but not dollar store scissors either.) the cut may be a little jagged, but the process of stretching fabric around and pulling it tight hides a multitude of foam-trimming sins.