My daughter (5yo) and I have narrow ear canals, so wax can become an issue for us. I have used "ear candles" on myself and they're incredibly gentle and relaxing (more info below). However, my daughter didn't want to do them because you light them, and she was afraid of that, so we use an OTC removal product-- as someone else said, they're kind of oily, so have a rag on hand. Basically, you have her lie on her side and put a few drops in. Lie there for 5-10 minutes, then switch and do the other side (I put a rag under the bottom ear when you switch sides). My doc also recommended that as we're rinsing her hair in the tub, just get a little water in her ears and for the most part, that has helped keep the wax from building up.
I've heard several people say that they've used the earwax candles on their children and the kids were fine with it-- I personally think they're even more gentle than the drops, since you don't feel anything with the candles (drops feel like water in your ear). You can get them at a health food store. Basically, it's about a 12" candle-shaped tube. You punch a hole in a pie pan, & put the candle through it (this catches any ashes). Lie on your side, put the tapered part in your ear and light the other end. Obviously you need someone else to light it & supervise. Once it burns for a few seconds, your helper blows it out. It continues to produce smoke, which goes down into the candle (not up in the air) and somehow draws the wax out of your ear and into the candle. It's actually pretty cool, and as I said, very relaxing. You don't feel any suction or anything.
If you want to use the candles with your daughter and think she might be afraid of the flame, I'd recommend trying to light it and blow it out before you put it in her ear (maybe out of her sight). I assume it will still work correctly, but I'm not sure. The flame may be what gets the suction going.
I'm surprised that your doc said earwax was dirty-- from what I understand, it's actually what helps keep your ears clean and protects them from infection. It's only when you get a build up that causes discomfort or hearing problems, otherwise it's healthy and normal. (So you still might be in the running for Mom-of-the-year!) Good luck! Let us know what works for you.