My Grandmother is a German Citizen that holds permanent residence in the US as she married my American Grandfather. I know some countries offer their passports to children and grandchildren of its citizens. I am American citizen but would be interested in getting an EU passport. Would it be possible to claim a German passport for myself(born 1981) since my grandmother is a German citizen?

No, you or your children will not be able to get German passports. If your grandparents did not submit a request of German citizenship for your mother, (and she was born in the US), she isn't German, which would leave you with no ancestral claim of German citizenship. You would be a German citizen only if your parent(s) were German citizens at the time of your birth, and registered your birth before your first birthday with a German Embassy.