Would it be ok to use a 9ft 14/3 Medium-Duty extension cord with a 1500 watt portable electric heater?
Yes, but only for temporary use (in that you have an eye on it). And make sure it is a halfway decent UL listed cord, not an imported knock off that has terrible connectors. Anything more than using the heater while attending the room, I wouldn't use a cord at all. Add an outlet where it is needed and run a new circuit. When connection to existing outlets, often normal branch circuits barely take the load, if any of the outlets are back stabbed (not using the screw terminals), be ready for loose connections to start burning apart.
The cord is UL Listed : 4-UL Certified from Home Depot. Only the heater will be used with it.
Again on the leaving it unattended, that would be my only worry. The space heaters often say "do not use extension cord" But that is mainly to keep idiots from using their 18 gauge lamp cord to extend it. The constant 1500 watt load is tough on connectors and adding a cord creates more potential for a connector to overheat and possibly start a fire. If your using it all the time, it would be best to add an outlet. If your using it to place the heater in a certain spot for temporary things, I wouldn't worry about using a cord.