One way would be to prepare all the colorizable elements of your illustration -- the regions within the black lines -- as separate white sprites (PNGs), with the ReplaceColor effect applied. Then when the user clicks in a "color palette" sprite, store the clicked color value as a global value, and assign this to the ReplaceColor effect of any clicked illustration sprite.
Depending on the complexity of your illustration, it could be a fair amount of work to position all the illustration sprites in the appropriate positions in the layout, but it should work.
There may be other options that treat the problem more like paint fills, but I'm not aware of how to do this.