Social Security has issued a statement on its Advocates webpage highlighting the level of its availability during the government shutdown which went into effect today, October 1.
Crucially, all SSI and SSDI payments will continue without change and on the same payment schedule as they have been. Therefore, SSI payments should have been made today, and SSDI payments will come without delay this month.
Your local SSA office will remain open with the same hours although you should expect longer wait times as there are going to be fewer staff to serve you. This means that you can still do all the things you could do last week, such as apply for benefits, request an appeal on a decision, get a nee SSN card, or even change your representative payee.
As has been the case for a few years now, the agency wants you to avoid seeing anyone in person if you can possibly help it. They are touting on-line services and the My Social Security account. You can set that up at my Social Security | SSA
Of course, the hope is that those in charge pull their collective finger out and get the government back up and running soon, but in the meantime, we should be only slightly inconvenienced, or so we are being led to believe.