There are really three options for filler sticks; shellac, soft wax, and hard wax.
Shellac is the traditional solution from the french polishing era, but it's still perfectly valid today, especially if you're applying shellac sanding sealer. The best way of applying it is with a heated spatula or a soldering iron set to a low heat. Hard wax filler sticks also need to be melted, and they're very similar in application to shellac. The best and most comprehensive hard wax repair sticks are probably the ones from Konig, but be prepared because they're not cheap! Soft wax filler sticks are the cheap and cheerful option, they're easy to apply but they're not as permanent as the previous two, plus on a chair I'd be concerned about them rubbing off on the sitter's clothes.
Many people will recommend sawdust mixed with glue. It sounds like it would be an ideal fix, but be aware that professionals hardly ever use this. The reasons are that sawdust is closer to end grain than long grain, so it soaks up the glue and looks darker. Plus it can be a pig to finish over a glue/sawdust slurry so it'll become even more conspicuous. You'll almost always effect a better repair by filling a divot with five minute, crystal clear epoxy. Just make sure it's one of the genuine crystal clear variants.
Finally, AJB Temple makes a good point, it sounds like you need some more practise cutting wedged through tenons before embarking on an actual project. Far better than a patch is a properly done job in the first place. I make quite a few pieces with wedged through tenons, and I've learnt the hard way that producing a clean and tidy joint, time after time, requires a fair bit of patience.
