EAxis would not be brave enough to make anything like that. Anything that is not a teen is IMMORTAL. Rule of the thumb and I am DEAD serious.
To answer your question - no, vampire kids of any age that is not a Teen will not burn in the sun. This is TL:DR answer.
Here are specifics: Vampire Babies are functionally exactly the same as humans. Vampire Toddlers are also the same EXCEPT for some CAS options being available to them (Eyes). Vampire kids are also more or less similar but again, there are some CAS options available and they are occasionally known to start hissing. Once a vampire becomes a teen they receive a special moodlet indicating that their vampire nature fully manifested, they are now functionally a vampire and have all vampire needs, levelling system and abilities and thus they CAN burn in the sun unless they unlock Sunlight Immunity ability later.
A word of advice (I know you didn't ask but I am going to tell you anyway) - consider having your kids read vampire lore books as they already can learn this skill as a child and this will help them in teen life.