Making animations in Blender

File > Export Mesh... and File > Start Recording... now support saving meshes as .PLY format, with vertex colors. Or you can use OBJ, which doesn't support vertex colors.

When importing a .ply file into Blender 2.8, use this option to show the vertex colors:

For animating meshes, the Stop-motion-OBJ add-on for Blender supports OBJ and PLY files. Use the same option as above to show the vertex colors in the viewport:

To see the vertex colors when rendering, make a new material and click here to set the material base color:

and then here to choose vertex colors:

Result:

(This was with the Eevee renderer. Cycles also supports the Vertex Node)