I think that would totally depend on the PNAM. If the PNAM knew that there was more than one group she was interested in, then she would have to make that decision about how to handle it. My cousin missed the intake period for her sorority, so she had to wait another year... when you consider the 25 years that passed from when she tried to join as an undergraduate to when she finally became a member, it's worth the wait.
Re: ruffling the feathers of other alumnae groups -- it's a possibility, but then, it's a possibility when you contact more than one group at a time. *shrug*
Edited to add: Woo-hoo! 1000 posts! 