I have two children (ages 2-4) who are just getting old enough to know when to make the worst possible face or simply run away when I bring out the camera.
Does anyone have any tips on how to get children to cooperate better for photography?
It is very highly dependent on the personality of the child. What works wonderfully for my daughter (Positive Feedback) may not work for other kids. So it helps if you already have a good handle on working with the child. With that said, here is what works for me:
My 3 year old is very appreciative of positive feedback. If I laugh at a funny face she will keep making the face again and again. If I tell her she's doing a great job of letting Daddy take pictures then often she will want to keep posing.
Give the Child Options
If there's one miracle parenting strategy this has got to be it. Every night: "Time to go to bed." produces "I don't want to go to bed." But "Do you want to go potty and then go to bed or just go to bed?" produces "Potty and then go to bed." So maybe you could try something like: "Do you want to hug your brother while Daddy takes the picture or do you just want to stand by him and smile while Daddy takes the picture?"
I recently took pictures of my daughter and a friend on a play date. It was easy for them to ignore the camera because they were more interested in playing with each other. As a result I got great shots of them chasing each other and playing together.
Engage in Pretend Play
Sometimes you can make the camera part of the game. Last week my daughter moved the bathroom stool into the living room, stood on it, and started singing songs. When we asked what she was doing she said she was on American Idol! So of course I grabbed the camera and tripod and my wife and I sat behind the camera and asked her name, what song she was singing and what show she was on. I got TOO MUCH footage and she still wanted to keep playing. Or maybe they're playing with a friend/sibling and perhaps you could direct the play a little bit by saying something like: "Are you going to chase each other?" "Do you want to play outside together?"
Patience, Patience, Patience
I also try to hang around with the camera for awhile and keep taking pictures (or pretending to) the whole time so she'll eventually get bored. Then I can potentially get some candids. Disclaimer: In theory this works perfectly. In practice I have two problems: 1) I get bored before she does. 2) By the time she gets bored she moves on to doing something completely different than what I was trying to get a picture of in the first place. (Which I suppose is fine if you're flexible.)
I would make the experience fun and exciting for them. Show them the camera, let them take a few pictures. If you let your children take a few photos here and there, they'll feel a lot more comfortable around the camera. They might even develop an interest in photography. You should answer the questions that they have and show them how it works. It might also help if you let them take a few pictures of you when you take photos of them. It also depends on your children's temperance. Maybe you could ask this question on a parenting forum (Do they have a SE for that?).