Nope as the door is 3/4" T&G boards. No way to mortise in a regular type lock. Only other thing you could do would be a surface bolt or hook & eye realistically.

Perhaps a magnetic latch mounted inside the casing with a knob surface mounted in the front? I did that with an old wooden medicine chest that didn’t have a functioning latch.

The wood is going to be a different color underneath the latch.

You could try House of Antique Hardware

or Renovators Supply