That's really weird. Keyboards don't really have different drivers, they've been mostly the same for the past thirty or forty years. Does the keyboard always work in everything else?
The problem might be something really stupid related to CPU usage -- NTVDM takes up 100% of your CPU time, always, whereas DOSBox is configurable to use more or less of it. What MIGHT (and I stress might) be going on is that the computer is not recognizing input from the keyboard because all the CPU is taken up by another process. But that just seems ridiculous, and if that's the case I highly recommend beating that laptop with a baseball bat until it's completely destroyed, then go out and pick up one that isn't retarded.

The baseball bat solution, while vaguely tempting, I feel would be bad idea as I paid $1600 for this tiny god just a few months ago.

I'm pretty certain that when I played Master of Magic in dosbox, the keyboard wasn't working even before I turned up the CPU usage.

Also, the keyboard doesn't stop working, I can still ALT-TAB, which is the only way to exit. I just can't do anything in the game, and keep running into Press Any Key to Continues.

I think you could try installing standard 101/102 keyboard drivers and check whether it works. It may happen that you won't be able to use additonal keyboard functions, but main buttons would work.. just a guess.. and you can also try switching to US language and US keyboard in regional settings.