If you're surprised, you cannot use Flash of Genius
Getting the more clear part out of the way first, the rule on "Surprise" state (emphasis mine):
If you're surprised, you can't move or take an action on your first turn of the combat, and you can't take a reaction until that turn ends. A member of a group can be surprised even if the other members aren't.
The rules on "The Order of Combat" state (emphasis original):
Determine surprise. The DM determines whether anyone involved in the combat encounter is surprised.
Establish positions. The DM decides where all the characters and monsters are located. Given the adventurers' marching order or their stated positions in the room or other location, the DM figures out where the adversaries are--how far away and in what direction.
Roll initiative. Everyone involved in the combat encounter rolls initiative, determining the order of combatants' turns.
Take turns. Each participant in the battle takes a turn in initiative order.
Begin the next round. When everyone involved in the combat has had a turn, the round ends. Repeat step 4 until the fighting stops.
Notably, surprise is determined before initiative is rolled, meaning that your Artificer is considered surprised and thus cannot take reactions such as Flash of Genius.
If you're not surprised, maybe you can?
This is honestly just unclear to me whether you can or cannot take a reaction when initiative is rolled (step 3 in the combat step-by-step mentioned above). We know you can take a reaction before your turn but this is also just a weird time in general, if it even is a time at all.
It is after the combat rules have begun (surprise and positions have been determined and established) and initiative rolls are ability checks so I could see an argument made that the Artificer can use Flash of Genius.
I could also see the argument made that the Artificer cannot use their Flash of Genius because they aren't reacting to anything. The section on "Reactions" states (emphasis mine):
[...] A reaction is an instant response to a trigger of some kind [...]
I'm not really sure what the Artificer would be reacting to here, though that comes down to the narration of what an initiative roll is more than anything else. The feature literally models the Artificer's "ability to come up with solutions under pressure" and what that ability can meaningfully be applied to is ultimately up to the GM. I don't see a lot of convincing details in this argument, but I can see the idea behind it.
I have not yet found any player character features that allow them to do something when initiative is rolled. The closest I've found are things like the Bard's Superior Inspiration feature:
At 20th level, when you roll initiative and have no uses of Bardic Inspiration left, you regain one use.
This just happens and does not use up the Bard's reaction, so though it isn't quite the same it does show that something can happen when initiative is rolled. Whether this includes taking reactions, is unclear for me and I have not found any rules explaining precisely when you are first able to use your reaction when not surprised.
That said, Thomas Markov did remember/find in their answer that the feather fall spell exists. A spell that explicitly allows you to take a reaction when somebody falls, an event that can certainly happen outside of combat. This is evidence that could point towards reactions being able to be used at any time and thus supports the idea that Flash of Genius can be used on initiative rolls.
I would rule that Flash of Genius works on initiative rolls
I let my players take actions, and bonus actions outside of combat so I see no reason not to allow them to take reactions outside of combat as well. Thus, I would allow my players to take reactions both inside and outside of combat; Initiative rolls must be happening at at least one of those times, so I would allow them to use Flash of Genius on initiative rolls.