Just doing a mental breakdown of precisely what is needed to make this:
The basic outside box is trivial for laser work. A bit of dremel or other action to get the bevel/fillet/whatever for the curved edges on the top. Spray lacquer for the finish.
Inside you have the four pieces that hold the rods to make the puzzle work. Those are where (as you mention) some glue will be needed. You could do an interlocking configuration to minimize this need. Or even do a tabbed insert to the side walls. Makes it a bit more apparent something is happening inside… but you can work the extra cuts and wood into a design, or cover with some veneer to hide everything.
The other option to avoid glue needs on the inside bits is to make them extend all the way to the bottom. Then it is press-fit, just like the Glowforge candle holder. This also keeps you from getting small parts trapped in the box.
You need to drill holes in the wood for the rods. No way to laser that (unless you keep the rods less than 1/2" thick I suppose).
The bottom of the lid you can do with laser by making it three stacked layers. That gets you the groove for the pins to slide in to. But it also makes the lid weaker, and prone to breaking when some meathead decides more force IS the answer.
So, if you want to keep the lid as a single piece, then you will need a CNC, table saw, or some other manner of making the full length groove.
The pins are most likely able to be purchased. But if you have to make some yourself, probably the easiest way would be getting some nails and chopping them down to appropriate size. You just need a small flared end to keep the pins from falling out when the box is open.
You could modify the design in many ways. Like using longer rods on a swivel-hinge, such that every corner has the hinge for a different side. Same end result, but now it only opens if you rotate in the proper direction.