Can MMR Vaccination Be Taken While Breast Feeding?

I was born in 1977 and had all the necessary MMR vaccines. I had a baby 3 months ago (through IVF). I just took a blood titer test for measles and my igG was negative - no immunity. I don't understand why I have no immunity. Is it safe for me to take another MMR vaccine and is it any risk to my 3 month old(unimmunized) baby. I am a breastfeeding mom. Also, my mom, who was born in 1951 took the blood titer test for measles and her igG also came back negative with no immunity.
Is it safe for her as well to take an MMR and is it safe for her to be in close contact with the baby