Where does Vixen S02E01 fall in Arrowverse timeline?

You noted that Firestorm was present in 2x01 of Vixen. The trailer for Season 2 seems to indicate that the story is taking place over a relatively short period of time and reveals that other heroes are present (which is confirmed by this article). Using those characters as a guideline, we can better pinpoint when Season 2 takes place.

Jefferson Jackson is Firestorm

So it has to take place after The Flash episode 2x04 "The Fury of Firestorm", which is when he became it.

Ray Palmer (the Atom) isn't presumed dead

The Season 2 trailer shows us that Ray Palmer will join the cast. Ray was presumed dead at the end of Arrow Season 3 and rescued in 4x06 "Lost Souls". Since Jax is present as Firestorm, it has to be after that rather than during Season 3.

Strangely though, Ray seems to have no problem being out in the open, despite everyone thinking that he was dead. However, we do see that he was using the suit in the pilot of Legends of Tomorrow.

Mark Mardon (The Weather Wizard) is not in Iron Heights

Mardon appeared in The Flash 4x09 "Running to Stand Still" where he was arrested by the end and sent to Iron Heights. So it must take place before that.

Jax and Ray are not time traveling

The pilot episode for Legends of Tomorrow takes place at the beginning of the year 2016 (sometime after Arrow 4x09 and The Flash 2x09, both of which took place before Christmas the previous year), and began with them being whisked away on a time-traveling adventure. So it would either have to take place before that or at some point after they return. But we know it's not after because...

Laurel Lance is Black Canary

Laurel stopped being Black Canary in Arrow 4x18 "Eleven-Fifty Nine" because

Damien Darhk killed her

Legends of Tomorrow made it a plot point in 1x16 "Legendary" that this couldn't be changed through time travel. So we know that Vixen Season 2 doesn't take place after this.

Given all this, Vixen Season 2 must take place sometime after Arrow 4x06 and before The Flash 2x10

Since the in-universe dates roughly correspond with the original air dates, this suggests that Vixen Season 2 takes place in November or December of 2015.

But what about Legends of Tomorrow 2x02 where the team meets a previous Vixen and act like they've never seen her powers before?

In this episode, the Legends are in 1942 and meet the Justice Society of America, among them Amaya Jiwe. Neither Jefferson Jackson, Martin Stein, nor Ray Palmer comment on the fact that they have seen her necklace before and seem surprised by her animal powers. In fact, Ray Palmer calls her "Animal Girl" before she corrects him and says that in the field she goes by "Vixen". Ray doesn't react to the fact that her descendant, whom he met, used the same name (given by Cisco).

I suspect that this was originally a production error, given that this Legends of Tomorrow episode was filmed in August 2016, before Vixen Season 2 aired. However, in episode 2x13 "Land of the Lost", Ray reveals that he has met Mari (almost certainly referring to the events of Vixen Season 2), but he intentionally keeps this information from Amaya because he is concerned that if he tells her, Mari might cease to exist.

Ray: I care about Amaya's granddaughter.

Nate: You dated her…

Ray: No, I've met her. Look, eventually, Amaya has to go back to her home in Zambesi in 1942…and she'll have a daughter, and eventually a granddaughter who will inherit Amaya's totem, and she will become a superhero in Detroit. If you change Amaya's destiny, that threatens Mari's existence and the existences of all the people that she saved. Amaya isn't like the rest of us. We were recruited because we didn't matter to the timeline, but Amaya does. She's special. She has a destiny and so do her descendants.

Nate: How come you haven't said anything?

Ray: Because it's dangerous to know too much about your future. Especially her.

Nate: What does that mean?

Ray: It's a long story, but it ends with her village being destroyed and her granddaughter in foster care.

Nate: Ray…I gotta tell her.

Ray: You can't. This is the only way.

Presumably Dr. Stein and Jax have similar motivations for keeping quiet about Mari.