Would applying Ivy Leagues with no SAT 2s hurt my chances that significantly?
No in general and “it depends” in a few cases. Since you ask such a broad question, you can’t expect an exact answer.
You know that most of them accept the ACT in lieu of SATs/SAT2s right? Both Y and P explicitly state that the SAT2s are optional – without any “wink wink” implication behind it.
Some Ivy Leagues and similar tier schools (one that comes to mind is Stanford) say two SAT IIs are “recommended.” Since many applicants will have this recommended testing it could potentially hurt you to not have it.