I recently built an axe using this guide, although I am no expert the following two points may be worth considering.

Wood is no good for axes like this - it cant take the force and just splits, as I found out a couple of times.

Sprocket and chain do work I used one very similar to the one you are looking at (you can get them in the uk if you just ring up a scooter garage) and am pleased with the results - however I am thinking of changing to a geared system for two reasons, first the chain does come off quite a bit which is annoying and secondly it would pack more of a punch with a bigger gear reduction. From suggestions given to me I am looking at getting: Steel Spur Gear MOD 1.0 12 Tooth - Part No: 4710-012, Tbot Steel Spur Gear MOD 1.0 60 Tooth - Part No: 4710-060, Tbot Steel Spur Gear MOD 1.5 12 Tooth - Part No: 4715-012, Steel Spur Gear MOD 1.5 36 Tooth - Part No: 4715-036 (part numbers are for technobots).

Hope this helps