Yes, if your minor children are eligible for a certificate of citizenship, you can fill out and file Form N-600, requesting one, on their behalf. This assumes, of course, that you need to do so. In some cases, U.S. Citizenship and Immigration Services (USCIS) will send you the citizenship certificate automatically. We'll explain all of this below.
Anyone who has acquired or derived U.S. citizenship, meaning received citizenship automatically through parents who were or became U.S. citizens, qualifies for a certificate of citizenship. This is different from the Naturalization Certificate that a green card holder receives after applying to become a U.S. citizen.
For details on who becomes a citizen at birth to parents who were U.S. citizens, see the article, Automatic U.S. Citizenship for Children by Birth To Citizen Parents (Acquisition). For details on who becomes a citizen when their parents becomes a U.S. citizen (the technical name for which is "derivation.")
In any of the situations described in these articles, when a child becomes a U.S. citizen through parents, it is an automatic process. They do not need to apply for their new status -- but may wish to apply for proof of that status, using Form N-600.
Under current law, if a U.S. citizen petitions for a foreign-born child to enter the U.S. and receive a green card, and the child receives the green card and enters the U.S. before the age of 18 and will live in the legal and physical custody of the parent, the child becomes a U.S. citizen almost immediately after entering the United States. This happens often in the case of a U.S. citizen's stepchildren or adopted children. (But it doesn't work for adopted children who entered the U.S. before the adoption was made final, in visa category IR-4.)
In recognition of this course of events, USCIS usually sends such children a certificate of citizenship within around six weeks of their entry to the United States. If you don't receive one after that time, however, you may want to submit a Form N-600 for your children after all. Or another, cheaper option is to file for a U.S. passport.
As you'll see in Part I of Form N-600, USCIS anticipated that some of the people filling out this form would be parents acting on behalf of a minor child (minor being under the age of 18).
If your children are over the age of 18, they will have to fill out the form for themselves. Special exceptions can be made for children who are disabled or who are otherwise unable to fill out the forms, so if you have a disabled child over the age of 18 who wishes to fill out the N-600 form to become a naturalized citizen, either you, another immediate relative, or a legal guardian will legally be considered eligible to fill out the form on their behalf.
You'll need to fill out Parts 1 through 3 with information about your child -- his or her name, birthday, address in the U.S., and so forth. (Make sure not to enter an overseas address here if you're basing your claim on the child's entering as a green card holder before age 18 -- the child must be living in the U.S. with you in order to be eligible for U.S. citizenship.)
Parts 4 through 6 are where you will need to enter information about yourself. In part 7, sign your own name. You do not need to fill out the section for preparer's -- this is for use by lawyers and so on.
Read the instructions carefully for additional information on what documents to send in. Don't be confused about whose photos to send in. They should be photos of the child, because the photos will be used on the certificate itself. You must write the child's name and A-number (if the child has a U.S. green card) on the back of the photo.
If you have questions or concerns about any immigration paperwork, it is important to consult with a qualified lawyer as soon as possible. Your attorney can assist you in confirming the child's eligibility for citizenship and completing the forms and documents required by USCIS.