On the site there are a list of questions with answers. A child has to be 16 to apply for one but you as the parent would apply, see below -
You should apply for an EHIC in your name and include your child as your dependant. You will each receive your own EHIC. You can apply for the card free of charge on the official EHIC online application route. Other unofficial websites may charge you if you apply through them. Alternatively, you can apply by calling 0300 3301350.