Does it matter what kind of airplane I fly?

Yes, and no.  Air Race Classic Rules spell out what airplanes are eligible to compete: “Airplanes are limited to STOCK and minimally modified STOCK airplanes, certified in the normal or utility category with normally aspirated engine(s) of not less than 145 horsepower, nor more than 570 horsepower, rated for CONTINUOUS MAXIMUM OPERATION”.
It is important that your airplane can fly a complete leg without needing to stop for fuel, taking into consideration that racing uses considerably more fuel than ordinary cross-country flights. More than anything, you should be well acquainted with, and comfortable flying, your Race plane. For more information, contact airplane@airraceclassic.org