To keep my answer specific, I'd probably say the biggest three factors in any personalized teacher would be (1) flexibility in the material (2) compatibility with students (3) inventiveness in the online classroom.
For 1) since the teaching will be personalized, the teacher would need to know the material well enough to interact with spontaneous questions, and being able to cater the pace to how quick or slow the student. For the age range, it seems like early "middle school" (and so I suppose beginning algebra? Like introduction to variables and equations and solving them?), so you'd also need to have a tutor who is experienced with handling those kind of questions.
(Speaking from my own experience, although I could do middle school math, I can't really teach it since I easily run out of ways to relate ideas to the student. So knowledge, as important as it is, is only as good as being able to communicate topics to someone new to it. So I'd probably look for experience in the math topics that you're looking for.)
For 2) you'd probably just have to get lucky with finding someone who works well with not only kids but also your kids. Teaching style might have something to do with it, but also just the general atmosphere when doing the class. Are the tutors primarily in it for the money? The work experience to get a different job? That they like kids and want to educate young minds?
(Again, for myself, when I've tutored, it's usually because I like communicating about problem solving with other people. I like how both student and teach and learn about the topic.)
For 3) since it's online, you'd need to somehow see how much set-up they have for online materials? Do they have pdfs of worksheets? Videos to watch? Tablet with pen to sketch or are they really good with a mouse and Paint? If they've ever had technical difficulties, then what happened?
Also, since you're looking online, would this person know your local school system so they know what kind of math will be happening in your children's school - or is the hiring for general mathematics knowledge, and so international tutors could be an option?