What has being a member of a Service Org. have to do with anything?
I am a proud member of APO, but at an other school, stated a Local which affiliated with LXA.
Maybe on the same Campus, that might be tuff.
As a Senior, yes it would be very tuff to get a bid. Most Orgs. want someone who will be there for at least 3 years to help with the Chapter.
Unless you can find one who is willing to say what the Hey, then, Your best bet is to wait for AI. But laying the ground work before hand would not be bad.
If you know a Soroity that you would like to affiliate with, explain to them that you know it would be tuff for them to Initiate you but, you would like to be with them for a future reference upon graduation. Who turns down help, especially from a person with a smiley face and a hand out pushing for someone who you are not officially with.