Your grandmother could be entered as a dependent on your mother's tax return if she receives either a Social Security number or applies for and receives an ITIN from the IRS before the due date of the 2019 tax return and she meets the requirements under the Qualifying Relative rules.
How to apply for an ITIN - https://www.irs.gov/credits-deductions/individuals/how-do-i-apply-for-an-itin
IRS Form W-7 - https://www.irs.gov/forms-pubs/about-form-w-7
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 $4,200 (social security does not count) in 2019
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.