can a grandchild living with you for the year under 21 years of age be claimed as a dependent

They may be able to be claimed as a dependent under the Qualifying Child rules if they are under the age of 19, have lived in your home for over 6 months during the and they have not provided for over one-half of their support.

To be a Qualifying Child -

1. The child must be your son, daughter, stepchild, foster child, brother, sister, half brother, half sister, stepbrother, stepsister, or a descendant of any of them.
2. The child must be (a) under age 19 at the end of the year, (b) under age 24 at the end of the year and a full-time student or (c) any age and permanently and totally disabled.
3. The child must have lived with you for more than half of the year. Temporary absences while away at college are considered living with you.
4. The child must not have provided more than half of his or her own support for the year.
5. If the child meets the rules to be a qualifying child of more than one person, you must be the person entitled to claim the child as a qualifying child.
6. The child must be a U.S. citizen or U.S., Canada or Mexico resident for some portion of the year.
7. The child must be younger than you unless disabled.

She can be your dependent if she didn’t earn more than $5,050 and you provided more than half of her support. I assume that she is not a full time student. 

Since she is age 19 or older then she may be claimed as a dependent under the Qualifying Relative rules if her gross income for the year is less than $5,050 and you have provided for over one-half of her support.

To be a Qualifying Relative -

1. The person cannot be your qualifying child or the qualifying child of any other taxpayer. A child is not the qualifying child of any other taxpayer if the child's parent (or any other person for whom the child is defined as a qualifying child) is not required to file an income tax return or files an income tax return only to get a refund on income tax withheld.
2. The person either (a) must be related to you or (b) must live with you all year as a member of your household.
3. The person's gross income for the year must be less than $5,050 (social security does not count) in 2024
4. You must provide more than half of the person's total support for the year.
5. The person must be a U.S. citizen or a U.S., Canada, or Mexico resident for some part of the year.
6. The person must not file a joint return with their spouse with the following exception -
You can claim a person as a dependent who files a joint return if that person and that person’s spouse file the joint return only to claim a refund of income tax withheld or estimated tax paid.