If light traveled at the same speed in raindrops as it does in air, could rainbow still exist?

No there would be no refraction and so the light wouldn't split into a spectrum. Refraction occurs when the speed of the wave changes usually on changing medium (usually due to density).

The previous answer is technically correct; however, there is more to it than just plain old refraction. You see in a rainbow forms as a result of total internal reflection (TIR). This is when light enters an object such as a fiber optic or a raindrop, and instead of leaving the other side, it bounces off of the edges a couple of times. Then as it exits it refracts separating the spectrum of colors. That is also why each person sees a completely different rainbow, so the next time you see one with someone else, tell them that yours is better.