Whether or not a persons name is listed on a child's birth certificate is not a factor in a child support case at all. His child support Order will be based on his and the Mother's income, with a deviation for him having shared physical custody. If he has been sharing custody for a 'few years' his paternity of the children will probably be presumed.
It would appear that your husband has already acknowledged his paternity of the children by filing for custody. That makes him responsible for sharing in providing for their needs, and, in his case, possibly paying support if that's the way their incomes line up.I suggest that he work with his ex to correct their birth certificates to reflect his name. The Bureau of Vital Records has simple procedures by which this can be accomplished. If they can't talk, or she won't cooperate, he should contact a local family law attorney to explore other options, the most likely being to file a petition with the court, seeking a court order that his name be added.
Disclaimer: Please note that this response does not in any way an attorney-client relationship between Kathryn L. Hilbush and the recipient. My responses are general in nature. They do not constitute legal advice. You are advised to consult an attorney regarding this and any other legal matters.