How best can I seal up an old coal door?
As you already know there is water getting into there, if you only seal up the inside to make it look better you will have water collecting between the door and your seal. You don't want a pool installed in your wall. 

To do this properly you really do need to excavate the outside remove the door and metal frame (if there is a steel load bearing piece across the top I'd recommend leaving that in and if there is a rain catch on the outside that could be left, but it doesn't look like there's either from the pictures). Patch the hole with material that functionally matches the foundation wall. Water seal the patched area then once it is dry back fill where you excavated. It would be good to make that area right against the seal a little higher grade when you fill it in than the area around it to help the water drain away and reduce infiltration.