Did the New York Stock Exchange ever close on a weekday so they could file paperwork?

Yes, from June 1968 until December 1968, they closed the NYSE every Wednesday so they could catch up on paperwork representing billions of dollars in unprocessed transactions. Even after the NYSE re-opened on Wednesdays in January 1969, they still had to close it early at 2pm for seven more months.

Forbes has a description of this:

Not to be forgotten, though, is the Paperwork Crunch. In a day of email and the Cloud and trading completed in microseconds, the idea that Wall Street needed Wednesdays off in the late 1960′s to catch up on back-office tasks seems especially quaint.

Yet, in 1968, the NYSE found itself sitting on more than $4 billion in unprocessed transactions. Trading had risen to 21 million shares daily; by contrast, even in the heavy volume days in 1929, trading never went above 16 million shares. Papers stacked on desks. A (now old) joke formed: If a fan blew the wrong way in a Wall Street office, visitors below could expect a ticker-tape parade.

“Everybody agreed that the securities-processing system had virtually broken down, and the only major point of dispute was who was more responsible for the mess: the back offices of the brokerage firms of the stock-transfer agents,” Securities and Exchange Commission Commissioner Ray Garrett, Jr. said in 1974. Some 100 broker-dealers failed, crumbling under the pressure of fulfilling those back-orders. The fix: an organization akin to the FDIC, the Securities Investor Protection Corporation. Wall Street would stick to the shortened weeks from June to December; in January, Wednesday trading resumed, though it ended early at 2 for another seven months.

Yes. To give brokerages the time to process paperwork, the American stock market, did stop trading on Wednesday's during 1968. Industry leaders first implemented the four day week around June 6, 68. https://www.newspapers.com/clip/75932396/

Interestingly, the four day weeks were out of necessity due to holidays, etc. in Nov 1968. https://www.newspapers.com/clip/75940046/

In Dec 68, they were planning to go back to full time at the beginning of the year, but decided better of it as the day approached. They then decided to operate on a reduced schedule.

One interesting side effect of being closed Weds to handle the paperwork, was a pattern of record numbers of transactions on the following Thursday. https://www.newspapers.com/clip/75943028/