I go to one university (University of Illinois at Chicago). I live quite a bit closer to another (Northern Illinois University). I can't speak for what NIU is doing, although it seems like Illinois public colleges try to do the same things at the same time. UIC is delivering the big lectures online (anything with over 70 students, of which O-Chem is included) and has the labs and smaller classes on campus. I am not taking any classes with small enrollments so I will be online through fall.
At both schools, Organic Chemistry I is offered as a separate lecture and lab. (You don't have to take them together and they are separate registration numbers.)
UIC's O-Chem I Lab filled up already for fall and I really needed it in fall 2020 for MCAT timing. Would it be okay to take the O-Chem I lecture at UIC and the O-Chem I lab at NIU?
These are unprecedented times. I imagine you will be fine taking it where it is available, especially being a 4-year university. If it ever comes up in question, just state what you stated here - that it was all filled up but you had an opportunity to take it nearby still in person while in org 1 lecture at the main institution. Just my opinion on it here. I would not fret it.