Does a bit in Flash memory store net charge
In the whole transistor, no. In the floating gate, yes. In the body of the semiconductor, yes.

Explanation time: Your understanding of capacitors is incorrect. Capacitors don't move charge. They have no field producing components and are completely passive in every way shape and form. They do store charge, of an equal and opposite kind, on each plate. Just like this, Flash also stores charge, however it does so on a floating plate by tunneling electrons onto the plate. The total charge in the universe is conserved because an equal amount of charge develops in the semiconductive material allowing for either a channel to be formed or destroyed. 

That "net charge" does exist, but only if you look at one plate at a time. And just as was discussed in the other question, eventually those charges will leak and the "net charge" on each plate will be gone, obeying Kirkoff's laws, which at its most simple form is really just conservation of matter (you didn't make any electrons from nothingness to charge the plate, you took them from somewhere). 

In the end it's just a matter of perspective: Do you include both plates in your arbitrary geometry used to measure field? If so, then there is no net charge. Do you include only one plate in your arbitrary geometry used to measure filed? If so, then there is a net charge creating a field going somewhere outside of your geometry.

Edit: The question was changed and so the answer has changed somewhat too.


  Now, the question is: is there net electric charge stored on the Flash chip in the form of electrons from the programming current trapped on the gates, such that Kirchoff's current law is temporarily breached (to be balanced when erasing the chip: charge conservation of course always holds), or are the charges of the trapped electrons balanced internally to the chip by, for example, holes (in the sense of semiconductor theory, an absence of an electron in the valence band) in the substrate?


The answer to which is the second option. Electrons trapped on the floating plate induce an accumulation of holes in the semiconductor. This accumulation layer acts like a break in the transistor circuit which is why it acts like a logical zero. As a reference, I pointed to a MOS capacitor, which is the basis of most field effect transistors in production today.