At one point we could influence its story, albeit it rarely (see the whole Jaques thing for how player actions changed the course of what Fdev had in store for him) and the other events that were mentioned. But not anymore.
Rarely is the key - and also largely unintentionally, since none of the players involved in the various crucial steps of how Jaques ended up where he did had that specific outcome in mind (some were trying explicity to achieve something else ... most were just seeing what happened if they did something potentially interesting).
That said - and with the note that it
is rare - "not anymore" is I think untrue, and the recent storyline seems to be deliberately taking account of player actions where practical, at if anything a higher density than historically found.
* The Alliance takeover of Witch Head after the end of the Enclave Initiative, while not entirely unpredictable, was certainly not inevitable, and only happened because of player action (through the in-game mechanics of the BGS). As a result, the current storyline has the joint AEGIS initiative largely abandoned by the Federation and Empire, and the Alliance facing/poking the Thargoids alone.
* The BGS position of the Marlinist colonies (and the efforts of at least one player group formed to influence this) have been directly noted in Galnet.
* The adoption of Hadrian Duval and Nova Imperium by an organised player group led to the Federation suffering a shock defeat (and reading between the lines, one which slightly surprised Frontier's writers too)
* Several other CGs and parts of storyline have followed on from others in a way which couldn't have happened had the other side won a previous one [1]. I'm sure that Frontier had plans for both possible outcomes, of course, but minor things like "does the Federation have an aggressive pro-corporatist movement?" or "how openly can the NMLA operate?" are still being influenced.
* While the odds are against it due to the sheer size of the search zone, it's certainly implied that someone
could stumble across the Scrivener Dredger "before" the next stage of that plot (and it may be that someone
does need to for the next stage of the plot to start, though that feels less likely).
[1] There's a definite change of pace here between "Act 1" of the current storyline (between the first NMLA bombs going off and the Sirius summit being interrupted by the Nine Martyrs' attack) in which player actions would largely have led to merely cosmetic changes if CGs had gone the other way ... and the current second act where they also lead to minor changes in direction and future events.
From what they've said the consequences of collective player actions are likely to get larger as the story progresses.
).
