It is clear that the Don used tachankas quite a lot. We have witten accounts of them, and the cavalry units all have MGs in their orders of battle.
But clear pictures of them, which we can definitively place as Don, are very rare. Fortunately, the British Imperial War Museum has a couple taken by their military mission to South Russia.
The first clearly shows that some MGs were carried on two-wheeled military carriages, which Russians called dvukolki. This is not my area of expertise, but these look like British ammunition wagons. The British shipped a lot out to Russian, and they are practical and sturdy. (They are also possible for gamers to find.) These may have been used by the Don infantry, as their cross-country capacity is far worse than sprung carriages.
White Russian Maxim machine guns mounted on ammunition carts
The next photo is a more standard Russian carriage, three horse I think.
White Russian troops with a Maxim machine gun on tachanka mounting, August 1919