Is your Alpha Gam friend a member of her local Alumnae Panhellenic chapter? If so, perhaps she can introduce you to ladies whose sororities have chapters at LSU?
I'm speaking for myself here, but I've always been against securing recs online from strangers. Normally I'd suggest that you contact your local Alumnae Panhellenic chapter first. I know there are tons of those in Alabama. You can get more information at
http://www.npcwomen.org . It's been done -- GCers writing recs for PNMs they don't know, but it's always been my personal opinion that you should at least make an attempt to contact your local alums first.
Now, speaking as an alumna who writes recs for PNMs every year, I like to get to know the PNM first. If I'm going to recommend this girl for membership in my Fraternity, she can't just look good on paper. She has to be well rounded and worthy of wearing my Badge. On the Alpha Gam RIF form, it asks the question of how well I know this PNM. I'd like to think that "I've known this woman/her family for x number of years/she was recommended to me by this person" pulls more weight than "I do not know this woman personally".