No, it's not possible to read the built-in text mode font data from "inside" the machine. (Update: Except perhaps in the Apple IIgs - see below.) In all Apple II models the font data is in a ROM chip used by the video output circuit.
In the Apple II and II plus (or europlus) the font data is stored in a "Text ROM" chip that contains only the font data. In the original Apple II the Text ROM is a General Instrument 2513 upper case character generator ROM. In Revision 7 the wiring of the ROM socket was changed to accommodate a 2316 ROM supplied by Apple, or a user-programmed 2716 EPROM.
In the Apple //e the font data is part of the "Video ROM" 2732 or 2764 chip that actually contains all possible video bitstream output combinations. (In the Apple //c, the same chip is called "CHAR GEN").
In the Apple IIgs (which is such a different beast that whether it really qualifies as an Apple II is debatable) there's evidence it can read the font data for testing purposes. From the Apple IIgs Firmware Reference, bottom of page 281:
C02C:00 246 CHARROM DFB 0 ; Addr for tst mode read of character ROM
As for third party Apple II video cards, such as 80-column cards, you would need to refer to the manual for a particular card, but in general they did not provide access to the font data either.
Here's the Text ROM in the Apple II video generation circuit on page 8-8 from Sather's Understanding the Apple II. Note that the ROM outputs don't connect to the data bus. Some of the above information about the Text ROM is from Sather's description of Text Video Output on page 8-15.