I'd get an HVAC consult. it's possible that your closet could survive without its own source of ventillation, as long as the rest of the bedroom has adequate registers. You'd want an expert to tell you.

8 months of planning and you hadn't noticed the air vent?

What I see also may be obstructing access to the attic access point, not a good idea.

I bet your fun at parties.

Is the third photo the back panel covering the vent? If it is, then you could remove the vent cover, cut a hole in the panel and then screw the vent cover over the panel opening. You could also spray paint the cover to match the panel so it is less noticeable.

Walk in closets are not required to have a supply vent, but having one is a nice comfort.

I'm not sure it's a supply register , looks more like a return grille cover to me. Either way both can be transitioned to a work like a register toe kick.

It's hard to tell, those could be moveable vanes inside the grill but the photo doesn't make it easy to see.

It could also be a grill for a jump duct.

Sometimes jump ducts are put into closets as a way to provide an air return path for a bedroom (the door of which can be expected to often be closed). If that's an interior wall and there's something similar looking on the ceiling on the other side, that's likely what it is. A peek into the attic would answer the question.

It looks like your scuttle will be in the way too. Why not just leave it all open and put covered boxes on top?