My maternal grandparents live in the US. They are a permanent resident living over there since 1990s. I need to go to the US since my grandfather is not in good health. Now the problem is that while they were applying for US visa there, they didn't mentioned the name of my mother in the documents or wherever it needs to be stated. So, as per US laws, they don't have my mother as their daughter. Is their a way I could apply for a US visa since Emergency Visas are only for the family members only.
As mentioned in the comments, if you have sufficient proof that you are related by blood to them and they are your grandparents, their not mentioning this in their earlier documents will just be considered an error. It can be considered a problem but certainly not for you.
Your birth certificate and other documents are perfectly valid for this purpose as well. Otherwise another way is to simply apply for a tourist visa and get a letter from your grandparents which says that you will be visiting them as a tourist. Make sure that you are able to fulfill all the necessary requirements for a tourist in this case then.