If I claim the child card credit does my mom have to claim it as income? She is on disability and doesn't file taxes.

If you report that you paid someone to care for your child, the IRS will want to see matching income on that person's tax return.

If you treat your mom as a household employee and give her a W-2, she would not need to pay taxes if her taxable income is less than $10,350.  If you treat her as an independent contractor then she needs to file a self-employed tax return.  She would not owe income tax unless the amount was more than $10,350, but she would owe 15% self-employment tax on her "net profits."

But if you treat her as a household employee, you will need to pay 15.3% employment tax in addition to giving her a W-2.  That's your share plus her share of medicare and social security tax, like you get taken out of your checks.  Since the credit is 20%, that's almost a wash.  (If you used an FSA, your tax savings could be as much as 40% so it might be more worth it.)

You might just want to let this go "under the table" as it were.