Can a standard NEMA 5-15 outlet be installed in a round ceiling box?
They do make receptacles designed to fit round boxes, for example a Leviton 1228 (I'm sure other brands have it as well, this was just the first I found when I googled Round Box Receptacle):



Depending in the screw holes on your box, you may also be able to fit a standard duplex receptacle and fit a wall plate designed for that purpose. These act as a wall plate and mounting bracket/adapter for mounting a standard receptacle in the round box.



Finally, if all you need is an ungrounded two prong plug, the easiest option might be to get some light bulb to socket adapter, like these GE 54276: