Is it possible to make a charcoal lump stop leaving marks?
There are a few ways you can do this.

You didn't say whether you wanted to maintain the natural colours within the coal or not, but if you do, you can use a clear spray lacquer, coat it in clear furniture varnish or any other coating that will seal the coal.

The suggestion by @BellaSwan of using  clear acrylic spray or nail paint will do the trick too but it will take a lot of nail paint bottles to coat a reasonable size chunk of coal.

If you wish to colour the coal lump, then you can use whatever kind of paint you choose; emulsion, eggshell, chalk-based, even gloss.