Yep I agree with PP's. Baby doesn't need anything but breastmilk or formula until she is a year. Your MIL is just plain wrong. However there's no point in trying to teach her how wrong she is. It's a double whammy: MIL and vegetarianism--don't even TRY to convince her, it will just drive you insane! Just do what you know is best and let her comments roll off of you. You can say things like "Hmm we'll think about that." and change the subject. Or "I discussed this with our ped she/he says baby is right on target nutritionally" (even if you haven't actually talked with your doc about it, you can still say this.)
As for your DC staying with your MIL, is this a needed thing? If so then you may just have to live with knowing that your babe is getting meat during those times. You could try telling your MIL that the doctor said no meat before a year--that might work. And If the sleepovers are not something that you and DH need to have MIL do, than you can decide to stop the overnights.
For what it's worth my MIL is very respectful of our parenting choices and my vegetarianism, and she STILL tried to feed my DD a huge spoonful of ground beef when DD was a baby! She didn't mean it to be disrespectful, she just wanted to share food with my DD. And meat eaters don't think meat is gross, LOL! So when she started shoving this big spoonful of meat at my little babe I was totally disgusted but she thought she was doing something nice! Different perspectives, you know? Anyway, I just calmly said "I'd prefer that she not eat meat until she's older," and MIL stopped.
Try not to sweat it too much. If it's not meat it'll be some other thing you are doing wrong. That's just the way it is sometimes with MIL's! :)