Just this morning, I started the car to go do some grocery shopping. I had no hot air coming out of the vents while in defrost. Is this a problem with the heater core?

 

Assuming all other variables were correct, probably. But they usually go out slowly. Was the engine warmed up, and does the heat not work in any position?

 

find both of the heater hoses that are goin into the firewall, after that (with car warmed up) grab the hoses
1. if one of them is hot and the other is cold then your heater core is stopped up
2. if both of them is hot and still no heat from any mode on your HVAC module, then i would say your heater core
3. if both of you hoses are cold then check something like a thermostat being stuck closed, or insufficient coolant in radiator

then tell me whats up with them if you need some help

 

Was there any air at all? Was it blowing cold air? The system defaults to defrost if there is a vacuum problem with the selector switch.

 

If you are getting hot air, there won't be a problem with the heater core, but either with the vacuum, controls or mode selection doors.