When Congress is delayed in passing appropriations bills, in the end, they always vote for continuing resolutions (CRs) that allow agencies to continue spending money at basically the same rate as in the just concluded FY.
House Republicans are working on a one month CR. The Washington Post reports that the CR they're working on would cut expenditures for FY 2024 by 1%. However, this pain would not be spread evenly. The Departments of Defense and Veterans Affairs would be excluded from the 1% cut. After these exclusions, the cut for other agencies, including the Social Security Administration, would be 8%. That would result in massive layoffs at Social Security. The agency would be largely incapable of functioning.
It's not completely clear that the votes are there for this draft CR in the House. It would certainly not pass the Senate or be signed by the President. It's not even a starting point for negotiations.
We're likely to see a government shutdown at the end of the month. This is actually the result desired by the small group of ultra right wing Republicans holding up things in the House. If your mindset is basically anarchist -- that government is so evil that we would be better off with no government -- then government shutdowns are a good thing.
If there is a government shutdown at the beginning of next month, most of the Social Security Administration will continue to function. The field offices, teleservice centers, payment centers, OHO offices, and the Appeals Council will not be affected.