Wall Street brokerage firms are staggering under a paperwork blizzard caused by heaviest sustained trading volume in history.

This blizzard isn't likely to bury any major firms, as one did in 1968 when trading volume was less than one third what it is today. Since then, stock exchanges and brokerage houses have installed sophisticated computer systems to process stock transactions.

But computers cannot work without people to feed them information, and the back offices of Wall Street are "filled with pooped people," according to the operations manager of one major Wall Street firm.

Some firms are having more problems than others. Paine Webber Inc. consistently is cited by its fellow brokers as having the most massive paperwork clog.

Paine Webber acquired Blyth Eastman Dillon early last month and has to cope with the sudden surge in trading volume at the same time it is trying to merge its trading and processing operations with those of Blyth.

Paine Webber has cut back sharply on the amount of trading it does on its own behalf to reduce some of the workload on its fatigued clerks.

"We've been laboring with it (the paperwork crunch). It seems to be improving, although I'm not yet ready to pound my chest," said Paine Webber Chairman James Davant.

Since the first of the year, more than 50 million shares a day have traded on the New York Stock Exchange, nearly double the daily average a year ago. Today Big Board volume climbed to 65.8 million shares.

Over on the American Stock Exchange, volume has averaged nearly 9 million shares a day, about triple the average of a year ago.

In the over-the-counter market -- where stocks are traded among brokers by telephone rather than on centralized exchanges -- volume has soared from a little more than 10 million shares a day during the first six weeks of 1979 to nearly 30 million shares so far in 1980.

Even Canada has felt the crunch of renewed investor interest in common stocks. Both the Montreal and Toronto stock exchanges have reduced trading hours -- a policy Wall Street adopted in 1968 but has eschewed this year.

Workers are putting in many extra hours of overtime, both in attempts to resolve questions about trades on the major stock exchanges and to match up trades between brokers in the over-the-counter market.

The New York and American exchange have ordered brokerage firms to come in on several Saturdays to resolve the mounting backing of so-called questionable trades (which can be caused either by the brokers who actually made the trade or by back-office paper processors).

The national Association of Securities Dealers, which regulates the over-the-counter markets, told brokers to work late three nights this week to clear up the backing both of questionable trades and of trades between two firms that haven't been "matched up" by their respective processing and sales departments.

Brokers say the problems are more acute in the over-the-counter market. When trades are made on exchanges, not only do the two brokerage firms have records, but an exchange employe records the trade in the exchange's computers.

"When people get tired, they make all sorts of mistakes," said Richard Ezra, who heads up operations for Drexel Burnham Lambert Inc. "And I've got a bunch of tired people who are making errors they normally don't make."

Nevertheless, Drexel Burnham and other firms that aren't in trouble "can continue to maintain control at 50 to 55 million shares a day," Ezra said. His back-office employes are getting out about two or two and a half hours late every night, he added.

Not only have brokerage firms had to cope with increased trading volume, they also faced a much-higher-than average rate of employe absences because of a citywide flu siege that New York officials say has reached epidemic proportions.

As a result, the classified ads here are filled with brokerage house positions for "experienced" back-office clerks, and some major firms report that they are being raided by smaller firms.

"We had a woman wire operator we had just given $20 raise to the first of the year. She seemed happy at $200 a week, but then she was offered $250 by another firm. We matched it. But when the firm boosted their offer to $275, she left us anyway," lamented the top operations man at a large brokerage house.