There was an article here on HN a while back about "Amish hackers", and one of the things I ended up learning from that was the Amish reliance on air compressors; compressed air was regarded as "Amish electricity" and would often be creatively applied to various (usually agricultural) machinery.
Perhaps something similar could be applied here? An air compressor powered by some renewable fuel (like wood gas) could pump air into portable canisters for use in power tools (or just be hooked directly into the tools themselves, much like in modern automotive repair shops) would solve this at the expense of some portability.
You can get portable air compressors. The thing is, it takes some energy to actually compress that air... which invariably comes from either an electrically powered or gas powered turbine. You could theoretically drive this by wood gas in a renewable fashion. In the building of my workshop, I'm considering air powered tools for this reason. Though, to some extent you could drive an alternator and generate electricity to run electrically powered tools. I'm not sure what a wood gas chainsaw would look like :P
In the above case, neither electricity nor combustion are necessary. The above example does rely on wind power, however (though the site claims that wind speeds as low as 8 MPH are sufficient); I'm not sure if there are any similar compressors that run on other energy sources.
Admittedly, some Amish communities do use diesel generators to power more conventional compressors, but I have a feeling that these would be easier to adapt to the posited post-apocalyptic wood-gas-fueled world.
Perhaps something similar could be applied here? An air compressor powered by some renewable fuel (like wood gas) could pump air into portable canisters for use in power tools (or just be hooked directly into the tools themselves, much like in modern automotive repair shops) would solve this at the expense of some portability.