I think you'd only be able to get a reliable science-based answer on this from someone like an industrial chemist, and it would have to be specifically tailored to individual types of plywood due to the range of adhesives used.
That said however we can make some educated guesses about how safe this might be in real-world terms. The first thing is, if you're in the US,
you are probably surrounded by board materials on a daily basis because of the building practices used there. This means your 'background levels' of any VOCs from plywood and other board materials would already be high, this is a fact of modern life and not something to be overly scared about and it puts into context what must amount to a tiny increase in exposure from occasional use of a solar oven.
In addition to this, similar cooking devices (including hot boxes, for passive cooking) have been made from plywood for many decades. In that time I'm sure someone would have noticed if there were a noticeable risk and the word would start to go around, and we know this is the kind of thing that would spread like wildfire and never go away — whether well-founded or not1.
Now don't take this as a guarantee of any kind, but I'd say this is likely safe enough that it should fall below your worry threshold.
If you are still concerned though, for security run the oven a few times before you use it to heat food. It's probably a good idea WRT the paint if nothing else. The coat of paint will benefit from going through a number of heating cycles, forcing it to release more volatiles as well as greatly speeding up the curing process compared to leaving it at room temperature. You can gauge the success of this by smell2.
 Two similar examples I can think of off the top of my head are MSG being responsible for "Chinese food syndrome" and cellphones being a fire risk at filling stations.
 Two samples of paint applied on the same day where one was heat-cured and the other was just left at room temperature will be in very different states on day three or four. The former should have nearly no odour while the second will still have that characteristic odour of fresh paint (indicating that VOCs are still being released).