To elaborate on StoodBridge's solution a bit...
You need to collect 200 salvage points for each of the original vehicles. You DO NOT have to fully upgrade the saucer piloted by Y the Alien.
Whammys spit out 2 salvage points.
Destroying a car normally is worth 5 or 6 salvage points.
Totalling a car (hitting it with a special or souped up weapon) is worth 9 or 10 salvage points.
When the salvage points show up, you have approximately 8-10 seconds to run over them to collect.
On each match end screen, it will show you how many salvage points you collected for that vehicle. As Stood said, if you need more after playing Quest mode, start up quick play with the vehicle you want to upgrade.