It's not so easy to make a list of who will be part of this project or not because we don't know in which circumstances the project was launched.
The first point is : Who lead this project?
Someone posted a few week ago a photo of a member of the Bioware Team with a T-Shirt. On this photo we could see "Pathfinder + Initiative + A.R.K.C.O.N".
Because of that, it's not impossible to think that project (Flying to Andromeda Galaxy) could be a "private" initiative à la Cerberus. As in the lore of the first games organization such as Cerberus exist, we could admit powerful groups with a large amount of ressources could exist and try to build such a huge project.
It could be also a project led by governments à la Macross Frontier where after the war with Zentradi, Earth authorities set up the project Macross to spread humanity through the galaxy to prevent its extinction on Earth (by an hostile alien invasion)
The second point is : When this project starts?
Many people tend to think this project started during ME3 and could be seen as a desperate measure to avoid total annihilation of the different species by Reapers. One good point for this scenario is that Bioware doesn't have to manage the outcome of the endings of ME3. And they could put anybody on board.
Another possibility could be after the end of ME3. Some organization (Council? Private investors?) is aware that species were on the verge of total annihilation so they set up a project to spread Milky Way civilization towards other galaxies. This scenario is a bit more complex for Bioware from a technical point of view. If you chose Destroy ending or Control ending it's easy. We could even imagine in the case of Control ending that this project is an initiative of a group who want to flee away from Shepard-God-Reaper.
The most difficult one is Synthesis ending. On a visual/technical aspect it would mean having a lot of character assets done in two versions (a normal one, one with glooming greenlight) or at least a way to apply some kind of green layer. This also raises questions about the relationship between species as with this ending Organics and Synthetics are supposed to have a mutual (perfect?) understanding of each other and I am afraid this is less interesting.
To manage your choices (Quarian vs Geth, Genophage cure?, etc...), Bioware could remove some ships/NPCs from the Pathfinder Initiative. That option would probably mean that Quarian and Geth would not probably play an important role in the next Mass Effect.
One last question could be : What technological breakthrough or event has allowed this travel?
Indeed in the current lore, travel between galaxy is not a technology which is mastered by Milky Way species. Common ground is that technology similar to wormhole could be a possibility. But it is also accepted that wormhole is a technical thing not a natural one, so what if the wormhole which allows this travel has been opened by ... a civilization from Andromeda? This could change a lot of thing especially if this civilization has "invited" some Milky Way species for its own agenda 
So I would say that any species we currently know about could be on the "ARK" according to what species Bioware wants to develop more.