Typical air pressure is about $15psi$, typical car tire pressure ratings are $\approx 30psi$. So the air will go from a pressure of about $45psi$ to about $15psi$, so the air in the tires will expand by a factor of something like $3$ (I'm assuming that the car is in shallow enough water that we can ignore it's pressure).
My ROM for the volume inside the tires is something like $\pi \times 1m \times 0.1m \times 0.2m \approx 0.06 m^3$ (one meter diameter, 10cm by 20cm cross section). Using all four tires gives us a compressed volume of $\approx 0.25m^3$.
Multiplying this by the expansion factor gives us something like $0.75m^3$.
The density of water is $1000kg/m^3$, so the overall bouyancy is $(750kg)\times g\approx7500N$. (The density of the air itself is about $1kg/m^3$ so I've ignored it in computing the buoyancy force.)
Although this calculation yields a result that gets to the order of magnitude, this result is too small by a factor of at least $2$ and probably $4$ to actually float the car.
Possibly under special circumstances, larger wheels and a very light car, this could be done.