How do I replace a mesh's texture with vertex colours?
You can sample the texture in your vertex shader and set the color appropriately. You don't say what system you're using, but for something like an OpenGL vertex shader, it would look something like this (this is a little old school, so it may need updating for modern GL) :

uniform sampler2DRect texture;
varying vec4 color;
void main ()
{
    color  = texture2DRect(texture, gl_MultiTexCoord0.xy);

    vec4    newVertex   = vec4(gl_Vertex.xyz, 1.0);

    gl_Position = gl_ModelViewProjectionMatrix * newVertex;
}


Your fragment shader would then just set the color of the fragment to the interpolated color coming from the vertex shader. Something like this:

varying vec4 color;
void main()
{
    gl_FragColor = color;
}