First, some good news. The Social Security Administration received its annual appropriation from Congress in September and that funds the agency through this February. Second, even if that were not the case, funding for programs such as Title II (SSDI), Title XVI (SSI) and XVIII (Medicare) are funded through a combination of taxes and long-term investments. The shutdown only affects funds annually appropriated by Congress. So the checks still get delivered.
Core programs like disability claims or appeals over benefits continue to function. But you would see a slow-down in non-essential programs. Employees for non-essential services would be out on a furlough, meaning they do not work and do not get paid. For Social Security, some non-essential tasks are issuing Social Security cards, applying for Medicare, getting benefit verification or correcting earnings records. Some 10,400 SSA employees will be furloughed out of the 63,200 total employees.
That being said, if you have an appeal pending in the federal districts courts, your action has been stayed. Yesterday, the Northern District of Georgia issued an order staying all cases involving federal government agencies. The Department of Justice requested that all federal courts pause proceedings in its cases, including meeting court-ordered deadlines. For those with appeals pending at the district court level, your case is on hold.