The courts have ruled that children do have some rights, but not all. It's based on their age and what particular right is in question.
A child cannot tell his government teacher to F-off and then expect the constitution to protect him or her. The child is not allowed to vote. A child is not constitutionally protected by the Equal Protection law compared to an adult. They are protected by Equal Protection in cases that involve all children. A child cannot pursue a lawsuit because they can't drink alcohol but adults can.
Therefore, no, a child does not have a constitutional right to buy and own a firearm.
What this boils down to is what the 1st and 2nd amendments say and do not say. Children have the same freedom of speech protections as adults. Just because there are consequences to what kids say, it doesn’t mean it is ever illegal for them to say them. They’ll never be charged with a crime for saying anything. Adults have the same problem. Just see what happens to your job if you tell that hot new intern she has a nice ass right in front of your boss.
Common sense law prevents kids from purchasing guns despite the fact that the amendment says nothing about children. If you gun nuts have the mentality that any current gun control laws proposed by lawmakers are unconstitutional, then your logic would have to apply to kids being legallly able to purchase and own firearms.