Yes, BUT new applications for SS disability benefits do not get processed; I am not sure about SSI claims, since those are a little more formulaic. I’m pretty sure new claims for standard SS retirement benefits go through, but claims for survivor benefits may not, since those need to be examined.
Also, this particular shutdown will happen during the time when the opportunity to sign up or change Medicare prescription benefits is happening, and that may be stopped. People who were on benefits, but covered by some other insurance plan, and had not signed up for Medicare at all, and now need to, may not be able to.
And Medicare claims may be slow to process.