I agree with everything @alexw said. 
And I don't think you need to change your style too much in order to fit in with what you think children's illustration is. You'll want to make your sample be art that you're happy and enthusiastic about illustrating an entire book in.
I think your loveliest art is on the top row of your website - all those textures and flowing forms!
They would not be unwelcome in a children's book, though I'm sure it takes a whole heap more time! But maybe you could incorporate a bit of that in this piece, if you'd like to.
Here's one suggestion: at the moment the dog is looking straight at the viewer and doesn't seem to know the boy is there. Maybe you could turn his head to face the boy, or he could be looking at his sore foot, or at the stethoscope? Creates a bit more interaction.
Aha! I think I've finally realised what it is about those flat shapes that doesn't quite sit right with me: I think they contrast too much with the more realistic shapes of the boy and the dog. I know it's hard to make veterinary cabinets and tables interesting... Maybe you could make the dog and boy more flat colour if that's something you like?
In fact the dog almost looks like a photo, even more realistic than the boy. So maybe you could make the boy and background more realistic to match the dog, or vice versa?
(I like how you've got the boy and dog's heads framed by the cabinet behind them.)
I've gone down a bit of a rabbit hole, looking at other children's vet books. I'm not sure these are going to be any help, but I had fun "researching!"

I can't find the name of this illustrator (
) but I think it's great how they've got texture in even the flat things, fun details everywhere (those posters!
), and a dynamic perspective.

On the other hand, Catalina Echeverri has kept this one very simple, with enough texture and contrast to show what's important.
And this one's my favourite, with a similar colour scheme and perspective to yours!


Everything in the background is desaturated purples and grey, with more saturated colours in the characters and a few props.
This is Paula the Vet by Julia Donaldson and Joelle Dreid and I recommend the read! I love how the vet gets more frazzled as the day goes by, and the silly surprise at the end.
Sorry that I've rambled on and on! I hope something here is helpful.
Have a fantastic weekend!