Eric's got a good point.
If you ask your alumni for money, the worst thing that can happen is they say "no".
For alot of the chapters that don't have alot of overhead, like houses, etc., alumni donations can help to reduce your dues.
From my past work experiences, if you don't ask for something, you're probably not going to get it.
So do a newsletter, or even a direct solicitation for a donation. It's easier than most people think, and pretty cheap.
I can recommend
www.48hourprint.com to use. Just design the piece, they print it up, and will bulk mail it at the cheapest rate possible. All you have to do is email them your list.
A thousand 5.5 x 8.5, full color postcards is $250, mailing services are $150 and about $.18 a piece for mailing your'e looking at about $580.
So, out of the 1000 postcards you send out, if you get 5%, or 50 back, with an average of $25, then you get $1250. A profit of $670. Not bad for an afternoon's work. If you get 10% back at $25, it's a profit of $1920.