If you are on DL operated flights, which it sounds like you are, than the rules of the operating airline are what applies to the flight. DL does not offer free same day standby. They have some kind of confirmed standby where you can be confirmed on another flight that day that departs w/in 3 hours of the original flight and there is a a fee to do so. I think it is $25 or $50. You can find more information on the Delta website or by reading the DL forum.
I can not answer your original question though as to whether you can use the program with award tickets.