I'm pretty sure that these days it's the newly-elected House that votes by state to determine the next President...
But was it always this way (ie before they moved the inauguration to January)? Was it the Congress elected in 1822 or the Congress elected in 1824 that chose J.Q. Adams?
The Constitution does not say. It provides for Congress to count the electoral votes, and if there is not a winner for the HoR to choose the President (and since passage of the 12th Amendment, for the Senate to choose the Vice President).
In 1788, after 9 states had ratified the Constitution, the Continental Congress set up the the schedule for the Constitution to come into effect. They chose the 1st Wednesday in March 1789, which happened to be March 3rd.
Congress under the Constitution first met on March 3rd, 1789 and took several weeks to achieve a quorum in both houses. As soon as they did, they counted the electoral votes and notified Washington and Adams of their election. Adams was able to get to New York City and was inaugarated first (and presided over the Senate). Washington took longer and wasn't inaugarated until June. So if Washington, had not received electoral votes from a majority of the electors, the 1st Congress would have chosen the President.
The Constitution doesn't provide much a schedule, so the 1st Congress defined one. Reasoning that terms were defined as 2, 4, and 6 years for representatives, presidents, and senators, respectively, and the initial term had begun on March 3, 1789; subsequent terms must also begin and end on March 3rd of odd years.
But this would present a problem with having a president determined before his term began. So Congress decided that the outgoing Congress would count the electoral votes, and potentially choose the President. So the electoral votes for the 1792 election were counted by the 2nd Congress in early 1793.
In the 1800 election, the electors voted on December 4th. Running Democrat-Republican running mates Jefferson and Burr tied. But the Federalists controlled the House (58 to 48), and many were opposed to Jefferson. But the state delegations were more evenly divided, and on 36 ballots between February 11 and February 17, 1801, the tally was Jefferson 8 states; Burr 6 states; and tied 2 states. An absolute majority of all states is required for election. Eventually the deadlock was broken and Jefferson was elected.
In the 1824 election, it was again the outgoing House that elected Adams. In that case he was elected on the first ballot on February 9, 1825, with 13 of 24 states voting for Adams.
It should be pointed out that Congress did not use a schedule like we are familiar with today. Typically, they met at the end of the years of their terms. The Constitution requires Congress to meet annually, with a default meeting date in December. Even if this wasn't specified, it was convenient, since it interfered less with agriculture, the dirt roads were frozen rather than mud, and the malaria and yellow fever were abated.
The schedule was like:
March 3 of odd years: term begins.
December of odd years: first session begins.
December of even years: second session begins.
If the second session ended at the end of the presidential term (every 4th year), the Senate would hold a short session at the beginning of their term to confirm appointees of the new President.
So the second lame-duck session was typically started after some of the representatives had been voted out of office, or decided to retire. Elections for representatives was not held on a single date. Some states would hold elections late in the even year, and it would be almost a year before they would actually meet be in session. Other states would hold their elections in the early part of the odd year before the new term began. Some states would hold their election later in the odd year, after the term began, but before Congress typically met. There were some special sessions in the spring/summer of odd years where some states were not represented.
A single national date for congressional elections was not set until after the Civil War.
The 20th Amendment moved the start of the terms into January and provided a 17 day stagger between the start of congressional terms and the presidential terms. It also set the default meeting date to the 3rd of January. The intent was that a newly elected Congress would meet at the start of their term, with a relatively short delay after their election, and also that the incoming Congress would count the electoral votes, and if necessary choose the President.
There is nothing in the Constitution that would preclude the Congress from setting a vote counting date (by the outgoing Congress) in late December, but this would be contrary to what was expected when the 20th Amendment was passed.
