1. What about going through a public health unit? (do you have them in Ontario? - in Alberta a public health unit is the place you go for baby & child vaccinations or shots you may have before you go somewhere tropical - travel shots, or where home care may be based - public health nurses who go to people's homes for assistance, or where speech therapy may be set up for kids who need it, or pre-natal classes etc). This is where I'd look first for the TB test in Alberta as detecting diseases that can be contagious is a public health issue.
2. If there's a particular volunteer agency that really wants your assistance - could they possibly cover the costs for your volunteer work - the TB test or fee if you have to pay for a police criminal background check? I wouldn't expect the volunteer agency to pay other costs - ie bussing or lunches - but if the TB test or other background checks are requirements it may be worth their while to help out the volunteers some. Ask - the worst they can say is 'no' - but if they'll cover those costs, its win-win for them and you!