After several hours of fiddling, I got this up and running.
It's pretty easy to do if you're not running on a Raspberry Pi / ARM processor (just follow the directions on minio.io after installing Docker on OMV).
If you're running OMV on a Raspberry Pi: Minio supports ARM, but the Docker version of Minio does not. To make up for this a few people have created Docker programs that just pull the latest ARM release of Minio. I'm using this one, and it seems to work well: https://github.com/pixelchrome/minio-arm
Then the only change from standard minio instructions is the docker program you run is called minio-arm.