As stated above, for your daughter to receive calls from people in South Korea they must dial as if she were in the US: 001 1 ??? ??? ????. They will also pay international charges. I'm not sure why people from the US haven't been able to reach her - make sure they know how to dial (some people don't realize that they don't need to add any international dialing codes; just have them dial her regular US number with no additions to it). Although I haven't heard of this plan allowing only family members to call someone carrying their phone in another country, it is possible that Verizon has this option. You should know better (calling Verizon might help). Also, I found the page below providing some dialing instructions for people roaming in SK:
At least for communicating within South Korea your daughter would likely be better off buying a local SIM card - this way locals would be able to call her without incurring international charges (she would probably pay less too). If she also wants to keep her US number she could get a basic low cost prepaid phone/plan.