Hey there!
Races are quite different from chases, actually, but you could absolutely use the system. If it was a two-participant race, I would just track their relative position, and if they're at Point-Blank range (which is where they would start at), let either participant attempt Close/Lengthen to pull ahead one range category.
You'd need to define a number of turns that the race will last, and possibly different terrain types for the course. Depending on the situation you might want to reduce the chance of obstacles (on a racetrack, for example).
For multiple participants it gets trickier. Everybody starts at Point-Blank range, of course, but then on each participant's turn, they (presumably) make a Close/Lengthen manuever, opposed by everyone else. Anybody who fails that reaction roll drops back a range category relative to the active participant, and everyone who succeeded at the reaction roll (thus preventing the active participant from pulling away from THEM). Each player's turn, therefore, triggers a set of reaction rolls (assuming Close/Lengthen maneuvers) from all players. You also have to keep track of every participant's range relative to all other participants. I would use some kind of board that has a row for each participant, place them all at the "front" and slide a participant backwards one space every time they fail a reaction roll on a Close/Lengthen. Hm. Not sure that would be huge amounts of fun, is my worry.
In any event, you might end up with a number of participants still at Point-Blank range to each other at the end of the race. A Reflex Save or opposed Drive/Ride checks might serve to determine a winner at that point.
First cut, no thought put into it, idea.