Your link isn't working, but I'm assuming you're talking about the shield-shaped badge with black enamel and the letters Alpha Phi Omega. I can't say I know what it is, but I can tell you what it might be.
According to Baird's, there have been 5 other fraternities (all locals) named Alpha Phi Omega. They were located at:
North Georgia Agricultural College
North Dakota Agricultural College
Ripon College
Wake Forest
University of Texas College of Mines
These were all active during and around the 1920s. So, the badge might belong to one of those groups. It also could belong to another local that was never noted by Baird's.
It would be interesting to find out if the back of the badge has any engraving on it.
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Kappa Delta
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