I don't know how much longer I can take it! Every time I text or talk to them and ask how she's doing, they say great. I was just over there again and IMO there's not ANY improvement. NONE. She's still as emaciated as ever, can't stand, can't hold her head up. At this point, you'd think there would be some change.
Any suggestions on how I can talk them into giving her to me? This just breaks my heart

put her on the scale and see what if any changes have happend. This will give you some concrete evidence instead of going of feelings. Is she staying at the vets?
If its a Vet, I imagine you could tell them you want to take her home as you feel responsible for her and that you thank them for their help in getting her stabilized and that would work.
If its just a family that has taken her in, I think if you could make something up like wanting her to be around other dogs if they dont have any or that you made a mistake giving her away and would like to know if you could have her back.
good luck L and keep us posted, I hope she gets better so she can be the getbig mascot!!!