Can my Mom apply for my married sister?

Only US citizens can file for their siblings. In your present status as a green card holder, you cannot file for your siblings. Next, your mother is only a green card holder herself. Under current law, she cannot also file for her married children. She must also wait until she becomes a US citizen before she can file petition for her married children. I write separately to advise you that even when you and or your mother becomes US citizens, those options are not viable. Assuming you are a green card holder, filing for your siblings [under Family 4th Preference takes at least 12 years for your siblings to come to the US. Next, when your mother becomes US citizen and files for her married daughter, Family 3rd Preference, your married sister will not come to the US for at least 11 years. There are options under immigration law through which, your remaining family members can legally come to the US in a much faster period. But you'd need to consult with an experienced immigration lawyer to help you.

Your mother can petition for her married son/daughter only after she is a US citizen; she can petition for any unmarried son/daughter even if she is a green card holder.

If your mother only has LPR status, she cannot petition for a married daughter. Your mother would need to become a US citizen before she can petition for a married daughter.