I can’t see any reason why the vehicle couldn’t be invented earlier, perhaps even with the very first machine guns like the Gattling or the Maxim (maybe the Nordenfelt?). I’d guess the main restriction on it would be seeing the role for the tachanka and having light enough machine guns that it can actually keep pace with the cavalry. Gatling guns could be towed by horses like field artillery IOTL (I assume you could do the same with a Maxim if you wanted to), so it really does look like the key is making someone see a need for a more mobile machine gun for cavalry units. Colonial conflicts seem like the best place to originate the idea.
On it’s impact, I don’t think widespread use of tachankas would in and of itself change the Western Front. Trench warfare, AIUI, is in large part of product of the force to space ratio which was dense enough to make any kind of breakthrough difficult because the defender was always able to reinforce their positions fast enough to prevent it. Sustaining and exploiting the attacks were the real problems, and I don’t see how a tachanka would meaningfully change that situation. That said, it might make cavalry formations more effective early on with a more mobile source of firepower.