Stealing those papers (from Congress' session) would not change the legality of their recording with the Archivist according to the Electoral Count Act (3 U.S. Code § 6)... which by the way says that states must send
six duplicate-originals of the same certificate under the seal of the State
I doubt all 6 copies were stored in the same location in Congress. The ECA partly details where some of the copies are to be kept:
the certificate or certificates so received by the Archivist of the United States shall be preserved by him for one year and shall be a part of the public records of his office and shall be open to public inspection; and the Archivist of the United States at the first meeting of Congress thereafter shall transmit to the two Houses of Congress copies in full of each and every such certificate so received at the National Archives and Records Administration.
I'm guessing that the ad-hoc Congress procedure would have simply involved soliciting new copies from the Archivist.
Thanks to Steve Melnikoff for pointing this out: the actual procedure that was followed in 2020 (and probably was the rule for while) is a bit more involved than simply captured in the law. As a NARA publication details the number of copies & duplicates is actually greater and their distribution is more "spread out", but it seems that only six of these actually have the vote counts:
The Certificates of Ascertainment list the names of the electors
appointed and the number of votes cast for each person.
- The States prepares no less than SEVEN originals, which are
authenticated by the Governor’s signature and the State seal,
and TWO certified copies. Alternatively, NINE originals may be
One original along with two certified copies (or three originals,
if nine were prepared) must be sent to the Archivist, David S.
Ferriero, c/o Office of the Federal Register (F).
The Governors must submit the Certificates of Ascertainment
“as soon as practicable” after their States certify election results.
- The remaining SIX original Certificates of Ascertainment will
be attached to the Certificates of Vote at the State meetings.
The electors record their votes on SIX “Certificates of Vote,” which are then paired with the SIX remaining original
Certificates of Ascertainment.
- After signing the Certificates of Vote, the electors seal
and certify the electoral votes in packages containing a
paired original Certificate of Ascertainment and original
Certificate of Vote. They immediately distribute the
paired certificates as follows:
- One pair of original certificates is sent to the President
of the Senate (Michael R. Pence)
- Two pairs of original certificates are sent to the Archivist, David S. Ferriero, c/o Office of the Federal Register
- The Archivist holds one pair subject to the order of the
President of the United States Senate in case the electoral votes fail to reach the Senate. The other pair is held
by the Office of the Federal Register for public inspection
for one year.
- Two pairs of certificates are sent by registered mail to
the Secretary of State of each State, who holds one pair
subject to the order of the President of the United States
Senate in case the electoral votes fail to reach the Senate.
- One pair of original certificates is sent to the Chief
Judge of the Federal District Court located where the
electors meet. It is held subject to the order of the President of the United States Senate or the Archivist of the
United States in case the electoral votes fail to reach the
Senate or the Archivist.
The President of the Senate and the Archivist should
have the electoral votes in hand by December 23, 2020.
If votes are lost or delayed, the Archivist may take extraordinary measures to retrieve duplicate originals.
As the new Congress assembles, the Archivist transmits
sets of Certificates to Congress, as requested. This generally happens when the Senate does not receive its set of
Certificates on time. The transfer occurs in late December or early January when OFR’s Legal staff meets with
representatives of the Secretary of the Senate and the
Clerk of the House.
As you can see, duplicates are held "at the order of" the President of the Senate (the VP) both by the Archivist and also by the States, so either set could presumably be sent on their order/request (the latter would have probably taken longer to fetch though). Further, there's a duplicate simply held for anyone to see at the OFR. The Senate sending some "mission" there to inspect and copy them would have probably been the last resort.