Will claiming a child tax credit this year result in a red flag for audit if I didn't claim it last year?

There would be no reason that claiming a child as a dependent that qualifies for the Child Tax Credit would cause any issue with the IRS regardless if this is the first time the child is being claimed as a dependent.

"I have custody. Nobody else has ever claimed him nor is eligible to claim him. I have not filed in several years since he was younger".

 Why hasn't anybody been claiming the child? 

A child can be the “qualifying child” dependent of any close relative in the household. If you live with someone else, e.g. your parents, they may have been able to claim your child.

Instead, you coulda allowed the non-custodial parent to claim the children.  Non-custodial parents are allowed to claim the child tax credit, but not the Earned income credit.

It may be possible for a qualified relative to file amended returns for past year tax credit. 

Since we have no information about the child you are asking about------are you his parent?   Or are you related to him?   You only say that you have "custody"----but we do not know how long the child has lived with you or why he has never been claimed as a dependent on a tax return before.   Without any of your personal information, if you would explain some of the the circumstances to us, we can be more helpful.   If you have been the custodial parent for several years and have not claimed the child, you have missed out on getting the child tax credit for 2021, and possibly missed out on receiving 2020 and/or 2021 stimulus checks for claiming the child---which you may still be able to get by filing tax returns for 2020 and 2021.