The US state where they were born keeps the records. You can order a copy of your child's birth certificate from the state if you are listed as parent and have a government issued ID card.
For example, Texas: http://www.dshs.state.tx.us/vs/reqproc/certified_copy.shtm
Other people asked questions on various topics, and are still waiting for answer. Would be great if you can take a sec and answer them