"I am a 51 year old American with a 9 year old daughter. Would I be able to bring my daughter with me if I got a teaching job in Korea?"
(I'm answering the question assuming you're a single mom. If you're married, your husband could home-school/take care of your child and none of the stuff I talk about below will be a problem).
Short answer: yes.
Long answer: yes, but I probably wouldn't do it for the following reason.
What do you plan to do for school? I certainly wouldn't put a foreign kid into a Korean school for a host of reasons including: large class sizes compared to Western countries (ie: no one will hold her hand), inability/unwillingness of anyone in the school to speak English to her, Korean being quite a difficult language to learn for an English speaker, lack of communication between you and the teachers and finally, the biggest reason of them all: bullying.
She will most certainly be bullied by her classmates because in Korea, people seem to seize upon any and every opportunity to assert their superiority over just about anyone who is perceived to be inferior, of which your daughter most certainly will be due to her lack of language skills. Teachers are also known to bully the weak and there was just a report in the news only one or two days ago about a teacher being disciplined for her repeated bullying of a mixed race kid. But, here's the thing: her punishment was less severe because she had won an award of some sort at some point in her career. Crazy. And certainly not recommended. Let it be indication to you that Korean parents are almost willing to give their left arm or pinky toe to get their kids out of the education system here.
Where does that leave you? International schools? Also a pretty terrible idea unless you're independently wealthy, which is probably not the case if you're coming to Korea to teach English.
Home-schooling? Yes, it's possible. But, what is she going to do while you're at work all day? Due to the language barrier, you'll have kind of an impossible time finding someone to look after her. And assuming you find someone who can speak English, it will just be way too expensive and impossible for you to actually make any money after paying for that.