Sable

I'll name the merchant you get the items from and their location, as well as the cost of each part.

I won't list each part individually because all 3 parts of each bike are sold by the same merchant (booster/engine, front, wings).

The best way to make a couple grand quick to start buying these parts is to spend 5-10 minutes farming Lightning Crystals at Crystal Plateau in the SW corner of the map. Grab the quest Lightning in a Bottle from the Machinist Zahra at Seven Sisters Station in the northern part of the SW region first. They sell for 100 cuts a piece. Make sure to keep 3 for the quest.

For stats, values are compared to the base bike (Gliding).
MS=Max Speed, A=Acceleration, H=Handling
BIKES
Nomadic Merchant Bobi - Burnt Oak Station (northern Sansee, 2nd area of the game)
Beetle Bike - 75 cuts each ... MS-2 A0 H+2

Boatmaker Jake - Up high, 1 grid square east of the cartographer's balloon in The Badlands.
Eyries Bike - 150 cuts each ... MS ? A ? H ?

Bike Merchant Ahvid - Fast travel to Eccria in Redsee (NW region) and he'll be off the platform to your right.
Giraffe Bike - 100 cuts each ... MS+1 A-1 H-2
Speedster Bike - 300 cuts each ... MS+1 A+4 H+2
Delivery Bike - 150 cuts each ... MS-1 A+1 H-1

Machinist Zahra - Seven Sisters Station in northern Hakoa (SW region)
Lighting Crystal Bike - 200 cuts each ... MS+1 A+1 H+1

Merchant Thiago - Marrow Bone Station in central Wash (SE region)
Nomadic Bike - 250 cuts each ... MS+0.5 A+1 H+1

Sarin - The Whale in southern Redsee
Finish the Historical Reconnection questline
Whale Bike* - ... MS+2 A+1 H0

Machinist Fir - Mysterious Shrine
Finish the An Ancient Race questline
Hicaric Ring Bike - ... MS-1 A+2 H+1

*Thanks to Calex dEUS for pointing this one out.