My niece came to live with me in June 2020. However, we did not get legal custody until January 2021. Can I claim her in 2020?

You may be able to.  Here is the criteria for claiming a dependent child. Your post indicates passing tests 1 and 3. If the other tests are passed then, yes you may claim her as a dependent.  

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 and younger than you (or your spouse if filing jointly), (b) under age 24 at the end of the year, a student, and younger than you (or your spouse if filing jointly), or (c) any age if permanently and totally disabled.

3. The child must have lived with you for more than half of the year.

4. The child must not have provided more than half of his or her own support for the year.

5. The child must not be filing a joint return for the year (unless that joint return is filed only to claim a refund of withheld income tax or estimated tax paid).