If there's no Secretary of X, then the Assistant Secretary of X fills the role.
Do Assistant Secretaries not also require Senate confirmation?
However, even if the Senate didn't push matters this far, it doesn't change the fact that Democrats and Regular Republicans between them command a veto-proof majority in both houses. Indeed, if the congressional races have gone the same as OTL, the Democrats
alone have such a majority in the House. With the cooperation of a dozen anti-TR Republicans, they have one in the Senate as well.
In short, you'd have an all but powerless Presidency - roughly comparable to the last two years of Andrew Johnson's administration. I think we may safely assume that the first business of the new Congress (if the old one hadn't already done it during the lame-duck session) would be to send an Amendment to the States setting a term limit on the Presidency, thus disqualifying TR from running again.
Of course, this is all very theoretical, as it depends on a virtually ASB situation. Wilson having a stroke would be no more likely to make his supporters vote for a Republican than Harding having one in 1920, or FDR in 1932, or LBJ in 1964, would have caused theirs to switch to Cox or Hoover or Goldwater. Either a new candidate would have been chosen or, if it was too close to Election Day for this, soothing bulletins about Mr Wilson's "temporary indisposition" would have been issued till the votes were safely in. Afterwards, once a solidly Democratic Electoral College had been chosen, it could be given out that he had had a sad relapse, and that "steps would have to be taken". But whatever those steps might be, they most certainly would
not involve the Presidency going to anyone other than a Democrat.