Did the Motorola 6847 have any memory onboard accessible to the user?
No it does not. It uses its 13 address lines to address up to 8 KiB of external display memory. In the MC-10 it uses part of the 4 KiB main memory. This is why most programs only use the basic text mode or 64x64x4 and 128x64x2 graphics modes. In fact, the higher resolutions (128×192x4 and 256×192x2) are only possible with a memory expansion.
to expand the meagre 4kb of the MC-10
That's what the 16 KiB expansion (Cat.Nr.26-3013) was meant for, bringing the system up to 20 KiB.
For those with a tight budget September '83 issue of Color Computer Magazine offered a modification stacking two additional 4016 2 KiB RAM chips on top of the existing and some TTL for decoding to bring it up to 8 KiB, providing the 6 KiB the high res modes need (*1).
It's important to keep in mind that the MC-10 was intended as direct competition to the Sinclair ZX-81 and Spectrum line (*2). So cost saving was top priority. That's why it had to go with a less expensive CPU than the Color Computer and no 6883 SAM (*3).
The Coco is like the full system design as shown on page 2 of the 6847 data sheet, while the MC-10 is exactly like the minimal setup shown on page 23 (sans the external character ROM that is).
On its own the 6847 VDG is a bit of an odd beast, as it does not contain any control/configuration registers. Mode setting is done by 5 dedicated pins, set in the MC10 via a 6-bit 74LS174 latch.
All timing is derivative of a single clock and memory addressing is fixed, starting from address zero and dependent on mode. That is why the video memory always starts from the RAM base at $4000.
In the CoCo a 6883 SAM - System Address Multiplexor - will do all addressing for the 6847 (and all other system components), essentially acting as DMA. It provides registers to move video memory to other addresses.
As mentioned, the MC-10 is intended as an absolute low-end device, so saving the 6883 and replacing the expensive 6809 and PIA(s) with a 6803 with an integrated port was a great step toward that goal.
*1 - And yes, there's a modern time 128 KiB memory expansion, called the MCX-128, as this video shows.
*2 - As well the VIC-20, still going strong in 1982/83.
*3 - Of course, all that saving, plus a long development time, meant that when the MC-10 was ready, in late 1983, it was too little, too late and way too expensive despite its 120 USD price tag). A larger memory might have changed this a bit, but not much ... then again, with 64 KiB, but maybe at a 199 USD price, it would have become a true C64 killer (which cost at that time 595 USD. Quite possible for a more simple computer with a production cost considerable below 50 USD. Except that would have meant for Tandy to take a risky shot - nothing the company was known for.