Rainbow Dash's best speed feat in base clocks in at about Mach 1.8k, the Rainboom allegedly doubling it to Mach 3.6k
Vegeta is roughly as fast as a Kaioken x2 Goku, as they would be the same power level, which would be 14% SoL or Mach 12.3k I believe.
Even if we give Dash her Rainbow Power, it would still only up her speed to about Mach 8.1k IIRC (Nigh-Sub Relativistic), which is still not fast enough.
Sure, in reactions she creams him (Mach 373k to Mach 12.3k), but in actual movement speed, Vegeta wins out, and would thus win a race.
In a straight up fight though Dash goes Rainbow Power and BFRs him to Hell and wins with her superior reaction and attack speed time.