Is it better to use vector or raster images for shapes with gradients?

MPEG 4 footage has an effective depth of 7 bit, so there is nothing you can do. It will always create bands as soon as you encode it, no matter how you treat it otherwise. And your artwork really has all the "Don't do this with compressed footage!" things in it - large uniformly colored areas or gradients spanning and exceeding the maximum 1024 discreet colors an 8 bit gradient can have with that range further being decreased due to the chroma undersampling as described. Unless you realyl completely rethink the design, there is realyl nothing you can do, even more so with 4k where even the tiniest glitch will be visible. You could probably get a reasonably decent encode if you settle on "just" HD, but anything beyond that is simply stretching the technical limits.

Mylenium

Nothing to do with vector vs. raster. This entirely depends on the project bit depth and that of the artwork, respectively. Parametric gradients created in AE can look just as band-y in 8bpc as can imported artwork and that kind of is the point: Importing AI artwork may not solve anything if you don't change your other workflows or adapt the designs to minimize banding. since you have not provided any actual info about your artwork or a screenshot we can't advise on how to improve things.

Mylenium