How to obtain nitrogen using only homemade utilities?
Crude process, but does the trick: You can use a combustion/oxygen consuming process, like a candle, to burn the excess oxygen. Eventually the candle will be too oxygen starved to sustain a flame, so some residual oxygen would be left (although this would already mean there is no fear of combustion of said tyres). Now to remove the remaining oxygen you'd use something like iron files (since you wanted to use only "household" stuff). Together with some humidity in that air, they will rust rapidly and should drive down the oxygen contents towards negligible. So now we have: 


a small trace remainder of oxygen, 
the equivalent of $\ce{CO2}$ from the combustion process, 
some impurities from the candle wax probably, 
trace gases such as helium etc. and ... mostly 
nitrogen. 


This already is not sustaining any combustion. But if you're a purist or want to use $\ce{N2}$ because it is lighter than $\ce{CO2}$, to remove the $\ce{CO2}$ you could use something like "roasted limestone", i.e. the stuff from you water kettle, which you heat so that you get lime, which then binds with the $\ce{CO2}$ to revert to limestone $\ce{CaCO3}$. Now you have something, I'd assume, like 98% nitrogen and surely to few traces of $\ce{O2}$ to cause any harm to your equipment. Of course, I'd buy bottled $\ce{N2}$, but for the sake of using household implements only, this might do the trick.