If the sticking point is that you require multiple materials on your chunks then I can probably add that in (just a case of having a material array instead of a single material and then passing that down to the generated chunks).
Other than that, this is more of a general shader question and I'm definitely not enough of a shader expert to advise how to do vertex animation like that.
You could perhaps use the alpha channel of the voxel colours to depict which vertices are water/lava in your shader.
All of the PicaVoxel PBR shaders are created with ShaderForge, so you can open them in ShaderForge and extend them with your own effects if you wish.
If you are able to create the effect you want and apply it to a standard Unity mesh, then I can advise further on how to apply it to a Volume.