I talked with the mom again when she picked her up Wednesday. I bluntly told her that she was very animated that day, telling me all sorts of stories, but I could not understand her. She just said "yeah, we just make her repeat it until she gets mad". The poor thing was trying to tell me what she wanted to eat and I could NOT understand her (nor could my DD!). I have said something to Grandma (who I have known for 20 years) and she said "I know, but they don't want to do anything". I am just going to keep repeating words, telling her to slow down and repeat for me, and hope they wise up. I only know a few words of sign language Carotte, but I think that would be overstepping because they don't know signing so it really would not help. It frustrates me, because we have had discussions about what I have gone through with my DD and she knows the programs I use for her but she just laughs off that her caretaker keeps repeating that she cannot understand her!! 
Given that you are a caregiver for this child, I just don't understand why it is you don't say to them, "I would like you to have her seen by a professional. I cannot do my job properly, because I can't understand what she is saying when she tells me what she wants to eat.
"Here is the name of the person I trust with this sort of issue. Or would you like me to call and make an appointment for you? Or would you be willing to authorize me to take her for the first evaluation?"
This is a serious issue.
You absolutely have standing here--you are the child's caregiver. And I would say that in a way, you *do* have responsibility. That's exactly why some people don't want to become babysitters or regular caregivers. Because that *is* part of the responsibility you take on.
Were I in your shoes, I might even be saying, "If you don't get her in to see a professional in one week, I will not be able to care for her anymore." Hoping that this sort of "yowzers, we're about to lose our babysitter!!" threat will get them to ACT.
And I'm not so sure that this isn't a form of abuse. I think that folks like CPS might actually intervene.
Oh, and you did too beat around the bush--You didn't say, "She needs to be seen by a professional. This is very serious, and you need to act."