I have no experience whatever of real upholstery, but have done a bit of "rough stuff" (e.g. a cloth-covered bed head after re-sizing it, and now a seat and back rest with foam cushion and "leather" cover for a child's pedal car). I started out with a big (-ish) manual stapler but found that either A) the staples (10 mm) wouldn't always drive straight, or B) you needed 3 hands, otherwise the recoil of firing the stapler allowed the staple to bounce back out a bit.
So I "upgraded" to an electric jobbie (B&D DIY model). Just as bad if not worse - exactly the same problems as above, especially re a 3rd hand to hold the machine hard flush against the job to prevent bounce on recoil.
Then Aldi (or maybe Lidl) had a cheapo air stapler and as I already have a fair-sized compressor with tank I decided to give that a try. Completely different world, and zero "bounce" problem. Really, the dog's doolies IMO. The only downside is the row the compressor makes, but for me anyway, that's a small price to pay for knowing that I can now drive fairly heavy duty staples reliably.
HTH
Edit for a P.S. Very nice looking chairs BTW. Certainly at my present stage of woodworking (non) "expertise" I wouldn't even dream of attempting such a thing. MAYBE (that's a definite maybe) I might have a go at such a project. Well done you. =D>