Per Grok:
The Congress can be vacated after the new legislators are sworn in on Jan 3.
"Precedents and Legal Framework: Historically, there have been instances where the House was without a Speaker at the beginning of a new Congress, but this did not delay a presidential inauguration. For example, in cases where the election of a Speaker has been contentious or delayed, like in 1855 or 1923, the House's inability to function did not affect the presidential swearing-in ceremony.

In conclusion, while the Speaker of the House plays a significant role within the legislative branch, their presence or election is not necessary for the President to be sworn into office. The President can be sworn in by the Chief Justice or another judicial officer, regardless of whether the House has elected a Speaker."