I took a look at the Davidson website, and I am massively confused.
For the women, it seems straightforward enough. It's very similar to the way dorm rush ran at my school. You visit all the eating houses, or just the ones that interest you most, and then you rank them and get matched to one according to your lottery number. The eating houses have no say in who joins them. (And no, NPC wouldn't touch this with a barge pole.)
It's the men's rush that confuses me. It looks like you have to receive a bid to be considered a member of the fraternity on the national level, but you can be a social member, and be considered a full member as far as Davidson is concerned, just by self-selecting. So the fraternities actually have to accept social members who are crammed down their throats?
As for Order of Omega - we had several "independent living groups" at my school, only some of which were Greek lettered. If I recall correctly, their members could be selected for membership in Order of Omega just like members of any fraternity or sorority, but few if any of them went out for it - I don't recall any ILG members being active members when I was.