I'm not quite sure what your situation is. Will your mom be home all day to watch the puppy as well as her own dogs, or are you wanting to just leave all of the dogs alone together?
Yes, she can be around your mother's dogs if they have been vaccinated. Just don't take her to any pet stores or parks or something where un-vaccinated dogs may be or may have been. But if there is not going to be anyone home to supervise, then I recommend a crate or puppy pen to confine the puppy while she is alone.