Can I be a low carb vegan? Short answer: No.
How about a low carb lacto-ovo vegetarian? Possibly, but really difficult.
Most of the low carb eating plans I’ve seen suggest that you can be a vegetarian or even a vegan. I frankly don’t think this is really realistic even for an adult. But for a child, a low carb vegetarian eating plan may be setting them up for failure.
There is a dizzying array of high carb food available–most of the food in the supermarket is high carb. The cereal aisle. The pasta and rice aisle. The baking aisle. The chips and crackers aisle. The cookie aisle. The bakery department. The juice and soda aisle. When you switch to low carb, your universe of acceptable food shrinks. That’s the bad news. The good news is that many of the choices are foods you might have wanted to eat but usually didn’t: pork bacon, steak, cheeseburgers. At first, the new eating plan is great. Bacon, again? Why Not? After a while, though, you have to get creative with your meal planning. No one wants to eat cheeseburgers three meals a day.
Now imagine the only protein and fat sources available to you are eggs, cheese and cream, and some tree nuts. (In my opinion, substituting highly processed soy products, like tofu and tempeh, for animal-based protein is not a good idea). Try to come up with 3 meals a day where most of your calories come from those fat and protein sources. You can eat nut butter, but remember you can’t spread it on bread. Yes, there are a lot of ways to prepare eggs, but probably not enough to keep your child on the eating plan.
My coach at my gym put it well while giving a nutrition talk. She said she likes animals, and she doesn’t really want to eat animals, but she needs protein and fat, so she does.