I think she would join, sort of, but they would be at each other's throats.
Irelia seems very reckless and straightforward in what she wants, and is not against extreme amounts of fighting or violence to accomplish her goals rather than by peace and negotiation. So it makes sense that she would see Riven as a means to an end, and their may even be part of her (the compassionate Ionian) that feels sympathy for Riven because she has experienced a similar feeling of loss, and may WANT to help Riven, even if she doesn't like her and disagrees with her views.
As for Riven, she would probably be hesitant because she still loves her home and doesn't want slaughter her own people, but on the flip side she is not opposed to "violence to end violence." The reason Riven and Irelia would have problems is because Irelia just wants to the kill the Noxian leadership, while Riven wants to fix her nation and bring it back it to the ideals and strengths that she grew to love. The choice would be hard for Riven because she would be forced to fight her own people, but doing so would mean possible redemption.
I honestly think these two would have some form of relationship lore-wise (probably not romantic, even tho I ship them hard af, pls rito) because of their backgrounds and goals, and because they share some parallels as foils (just like Riven and Yasuo are foils of each other), and Riot could write some very good stories about these two and what they do. I'm pretty sure that we are going to get an Ionia lore update like we did with Noxus before, and because there wasn't anything for Riven in the update, Riven will probably be included with the Ionian's and if she isn't, she will be a part of Yasuo's, and possibly Irelia's. I can't wait to see what happens.