I was thinking of using dowels to keep a few plywood panels together. No glue, no frame, no nails or screws.
That would work.
Dowels
Having taken apart some dowelled furniture which was unglued or too-lightly glued in the factory this is definitely a viable method of construction even for certain furniture items, so should be perfectly adequate for what you plan to make.
Automatic perfect alignment can be guaranteed if you use through-dowels rather than the more common method of drilling two separate holes so the dowels remain hidden.
Screws instead?
The basic methodology is the same, but screws would make disassembly and reassembly much easier (once you unscrew the thing falls apart by itself).
Screwing works best with carefully chosen screw types1 but can be done with virtually any screw if required2. You need to pre-drill clearance and pilot holes holes for the screws naturally, in addition to doing a countersink or counterbore for the screw head to sit into. Note: there are drill bits available which will do all three in a single operation, but doing this infrequently the multiple drilling operations aren't too tedious and saves having to buy a set of bits you may seldom use.
Now I've heard people say you shouldn't drill into the edge of plywood, because the glue would loosen and the fibre sheets would come apart.
Well if anyone says that without providing any additional info you might want to stop listening to anything else they say. The details matter, hugely. Not only can you drill into the edge of plywood but doing so has been standard practice for roughly 70 or 80 years!!3
but does it really happen by just drilling into it?
It can, yes.
Should I avoid drilling into the edge of plywood to put in some dowels?
As you expect by this point, nah! But there are some caveats. The quality of the ply is important (this is one of the bits of additional info that matter) as is the size of the hole in relation to the thickness of the plywood...... it should be obvious that there's little risk in drilling a skinny hole into thick material.
For larger holes, while you can do it with standard drill bits (twist bits) you'll often find that lip-and-spur / brad-point bits are the better choice.
Particularly where the holes will be quite large in relation to the plywood thickness (1/3 or larger), and also when using twist bits which have more of a wedging action, it's a good safety measure to firmly clamp the plywood at each hole location as you're drilling, or, clamp battens along the edge so you can drill an entire line in one go without having to reposition any clamps.
1 Such as Confirmat screws.
2 Although with basic screws if the requirement is to take the thing apart many times over the expected lifespan reinforcing the pilot holes would be advisable. This can easily be done by dribbling some glue into them. If using epoxy or superglue don't reinsert the screws until the glue has set unless you've waxed them thoroughly!
3 For two of the primary joining methods in plywood construction — glueing and screwing, and glueing and dowels.
Edit: I'd forgotten this Answer from 2016 but it covers a lot of the issues, What length screw should I use for a computer desk