And of course the contribution from the sidewalls (as a percent) will change as tire pressure increases.
On the other hand, if you had hollow steel "tires", the stiffness would be everything and the contact patch would change only imperceptibly with a change in tire pressure.
Car tires are obviously somewhere in this range.
A car? Not much. At any reasonable driving air pressure the air is doing most of the work.
A pickup/van with E or better rated tires? Eh, some but not much in proportion to the weight said tire is expected to carry, especially if the tires are anywhere near the top of their pressure range.
A skid steer or some other off highway machine with "bajillion ply" tires expressly designed to be resilient to sidewall punctures that runs low pressure because all of the suspension is in the tire? Yeah, sidewall stiffness accounts for a lot in that case.
Consider a section of tire wall, with the same pressure on either side. It is curved outwards, and if you put compression on it, it will buckle outwards. If there is higher pressure on the inside of the curve, that would help with the buckling, not prevent it.
In an inflated tire, as in a bubble, it is the tension along the curved surface that resists the internal pressure. What keeps the sidewall of an inflated tire from buckling is that the buckling would decrease the radius of the curve locally, increasing the net inwards force of the tension over the area that is buckling.
I have some data on this for a few different tyres but I don't think I can legally quote it here :(