The Chairman of the House Ways and Means Committee as well as the Chairmen of that Committee's Social Security and Human Resources Subcommittees have written President Trump to urge that he nominate a new Social Security Commissioner.
I think it will be difficult to find anyone other than a career employee willing to be nominated now. The term of office ends in mid-January 2019. By the time anyone could be confirmed, their term would almost be at an end. Confirmation to a new term thereafter might be difficult, especially if Democrats gain control of the Senate. That's unlikely but not out of the question.
There's also the problem that Social Security is wildly unpopular with prominent Republicans. It may be difficult to find someone interested in the job who wouldn't be insistent on attacking the benefits programs he or she is supposed to be administering. However, almost no Senate Republicans would want such an unpopular crusade.
Finally, there's the general problem that few people want to work in Donald Trump's Administration. It's an incoherent mess very possibly based upon near traitorous conduct, not to mention that Trump treats people dreadfully.
I'd suggest a recess appointment of a caretaker Commissioner. That can be done quickly without confirmation. It's good until the beginning of the next Congress. Of course, doing that would require that Trump have some interest in governing instead of just dispatching absurd tweets in response to Fox and Friends.