The AMA search goes by the residential address of the physicians. So if a physician lives in a suburb of Chicago such as Oak Brook, but practices in Chicago, he/she won't come up on a search for physicians in Chicago. Likewise, physicians who reside in a neighboring city like Edmond will not come up on a search for physicians in Oklahoma City.
Regarding residents, I don't know whether or not they're included on the database.
If you are looking specifically for osteopathic physicians, the AOA has a similar DO-only search that is probably better to use:
http://www.osteopathic.org/index.cfm?PageID=findado_main
If you search for Emergency Medicine physicians (primary specialty) in Oklahoma (without specifying a city), perhaps you'll get the results you're looking for.
Neither database claims to be 100% complete or accurate, but both should be pretty comprehensive. Both provide information for patients, but the info could also be helpful for those seeking physicians to shadow (or learning more about the background of the physicians they're shadowing).