The Dow Jones Industrial Average, which had jumped by 105 points since last week, cooled off today, falling by more than 9 points in heavy trading.

While the carefully watched barometer of blue chip industries slipped to 1,003.68, there were signs that individual investors have begun following the herd of institutions that led the stock market rally.

A record number of orders was handled on the New York Stock Exchange, the total trading volume of 127.7 million shares was the sixth heaviest ever and the stock market wire was delayed for more than an hour.

The decline in interest rates -- credited with spurring the market boom -- continued today as several major banks dropped their prime lending rate from 13 to 12 percent, the lowest since September, 1980.

Investors continue to pull cash out of short-term money market investments to invest in stocks in hope of better returns, analysts say. The rates for government-insured FHA and VA mortgages also were cut, from 13 1/2 to 12 1/2 percent effective Wednesday.

In Washington, President Reagan was preparing to deliver a televised "progress report" to the nation at 7:30 p.m. Wednesday from the White House, with live television coverage by NBC and CBS. ABC will not carry it.

Reagan, who had called Monday's leap in the Dow Jones average past the 1000 mark "a leading indicator of what's coming down the road," returned to his old skepticism about the importance of the stock market today. When told about the market's decline, he said, "I stopped counting" on Monday.

Reagan said today that the latest drop in the prime rate was "another burst of sunshine."

The president's Monday remarks came under attack today from House Speaker Thomas P. (Tip) O'Neill Jr. (D-Mass.), who said the Wall Street surge is not a true barometer of the nation's economic health.

"I'm not interested in what Wall Street is doing. I'm interested in Main Street," O'Neill said, adding that the stock market represents "a lot of money in the hands of a few."

There were signs, however, that small investors were beginning to jump on the Wall Street bandwagon.

"We had the biggest public participation day of the post-Aug. 11 rally, and that's why the tape is running late," said Robert Stovall, senior vice president at Dean Witter Reynolds Inc. There were more than 106,092 trades on the market today, up from 102,825 Monday. Each figure broke the previous high of 94,239 set in October, 1979.

While more individuals appeared to have entered the stock market today, institutions were taking profits, particularly from the blue chip stocks that had led the market's dramatic turnaround. At the same time, however, some bond prices jumped more than 2 points in heavy trading.

But market experts said they expect the turbulence and intense activity of recent days to continue for some time.

"There should be these peaks and valleys through at least Election Day," noted Charles Lewis, a vice president at Shearson/American Express Inc.

"There will be more violent action," said Newton Zinder, senior vice president at the E.F. Hutton Group Inc. "We're not ready for a big decline yet, but there should be no more straight up either."

Stock experts are predicting that, barring dramatic developments, the market is likely to pick up again late this week, when many stock options expire. More money from all savers certificates and money market funds is expected to move to stocks.

"There is only one place to put money: the stock and bond markets," Lewis said.

The market's turbulence was apparent at various points during the day. The Dow industrial average swung over a range of nearly 35 points, up as high as almost 1,027, and down as low as 992.00.

Dramatic movements like these and the market's recent boom as well have been prompted primarily by the expectations that the Federal Reserve Board will continue to support lower interest rates.

The drop in the Dow industrial average came as 931 NYSE stocks gained in value and 776 declined. The day before 1,501 stocks had turned up, while only 293 dropped in per-share price on volume of 138.5 million shares, the second highest on record.

Exxon was the big board volume leader, rising 1/8 to 30 3/4. But the technology stocks which had led Monday's surge fell off today, as American Telephone & Telegraph Co. fell 1 3/4 to 61 and International Business Machines Corp. dropped 3/4 to 82 3/4. Of the 30 major firms whose securities make up the industrial average, 19 went down today, while 10 went up. One was unchanged.

The American Stock Exchange index was up 2.91, closing at 307, with volume up today to 9.3 million shares from Monday's 8.7 million. The average price per share jumped by 12 cents. The over-the-counter market also was off slightly as the Nasdaq index fell 0.15 to 202.16.