Whether your child has a Social Security Number (SSN) or an Individual Taxpayer Identification Number (ITIN) does not impact whether they qualify your return for the Child Tax Credit. They must be a U.S. citizen, U.S. national or U.S. resident to be considered a qualifying child for purposes of the Child Tax Credit (and also meet the other qualifying factors). Having an ITIN does not prevent a return from being eligible for the Child Tax Credit (regardless if the ITIN belongs to the taxpayer, spouse or child).

In the TaxAct program, if you enter an ITIN you will need to select “Resident of Canada or Mexico” or “Other non-citizen or nonresident alien” as the Citizenship type in order to see the additional questions necessary for determining whether the child qualifies as a resident alien or nonresident alien for tax purposes.
 
Additional Information
Per IRS Publication 1915 Understanding Your IRS Individual Taxpayer Identification Number (ITIN), Page 20:

What are the rules for the Child Tax Credit/Additional Child Tax Credit (CTC) as they relate to an ITIN?
Although your child may be able to obtain an ITIN for purposes of being claimed as a dependent on your U.S. Federal tax return, your child must be a U.S. citizen, U.S. national or U.S. resident to be considered a “qualifying child” for purposes of the Child Tax Credit. Refer to IRS Publication 972 Child Tax Credit for additional information. 

Per IRS Publication 17 Your Federal Income Tax, Page 223:

Qualifying Child
A qualifying child for purposes of the child tax credit is a child who:

1. Is your son, daughter, stepchild, foster child, brother, sister, stepbrother, stepsister, half brother, half sister, or a descendant of any of them (for example, your grandchild, niece, or nephew);
2. Was under age 17 at the end of 2017;
3. Did not provide over half of his or her own support for 2017;
4. Lived with you for more than half of 2017 (see Exceptions to time lived with you, later);
5. Is claimed as a dependent on your return;
6. Does not file a joint return for the year (or files it only to claim a refund of withheld income tax or estimated tax paid); and 
7. Was a U.S. citizen, a U.S. national, or a resident of the United States. If the child was adopted, see Adopted child, later.

For each qualifying child, you must check the box on Form 1040 or Form 1040A, Line 6c.
 
Adopted child. An adopted child is always treated as your own child. An adopted child includes a child lawfully placed with you for legal adoption. 

If you are a U.S. citizen or U.S. national and your adopted child lived with you all year as a member of your household in 2017, that child meets condition (7) above to be a qualifying child for the child tax credit.

Exceptions to time lived with you. A child is considered to have lived with you for more than half of 2017 if the child was born or died in 2017 and your home was this child’s home for more than half the time he or she was alive. Temporary absences by you or the child for special circumstances, such as school, vacation, business, medical care, military service, or detention in a juvenile facility, count as time the child lived with you.

There are also exceptions for kidnapped children and children of divorced or separated parents. For details, see Residency Test, in chapter 3.

Qualifying child of more than one person. A special rule applies if your qualifying child is the qualifying child of more than one person. For details, see Special Rule for Qualifying Child of More Than One Person, in chapter 3.