If the Democrats have their way, one of the hottest issues in this fall's congressional elections will be the nuclear freeze.

Minutes after the House voted 204 to 202 Thursday night to support President Reagan's arms control policy, which allows for a weapons buildup during arms control negotiations, Peter Kostmayer's campaign manager was on the telephone to find out how James K. Coyne (R-Pa.) had voted.

Kostmayer, a former Democratic congressman from Bucks County, Pa., is trying to unseat Coyne, who, after days of indecision, voted for the administration-sponsored resolution rather than for a measure supported by the nuclear freeze movement.

Within hours, every newspaper in Coyne's district had a statement from Kostmayer that Coyne had "crumbled under political pressure."

At a Capitol news conference shortly after the vote, congressional proponents of the freeze resolution, sponsored by Rep. Clement J. Zablocki (D-Wis.), declared themselves "elated" despite their loss.

"We'll win at the polls in November," said Jonathan B. Bingham (D-N.Y.).

Moderate Republicans worried. "I predict the freeze will be the greatest single issue . . . in the 1982 elections," said freeze supporter Jim Leach (Iowa), one of only 27 Republicans to vote against the president. "The accountability on this vote will be dramatic, and it is being underestimated by most members of Congress."

Stanford E. Parris (R-Va.), a co-sponsor of the administration-backed resolution introduced by William S. Broomfield (R-Mich.), said the freeze was "the preeminent foreign policy issue" in his district. His opponents this fall--former representative Herbert E. Harris II, a Democrat, and independent Austin W. Morrill Jr.--support the freeze.

But Parris is counting on portraying the Zablocki resolution as a move for unilateral disarmament, although it called for "a mutual and verifiable freeze" by the United States and the Soviet Union.

"I think the majority of people in my district do not believe that this nation can ensure peace if we do not prepare for war," Parris said. "We can't lay down our arms and trust the Soviets to be gentlemanly."

In last-minute telephone calls to House members during the debate Thursday night, Reagan, Vice President Bush and a small army of other administration officials argued that a U.S. buildup of first-strike weapons is necessary as a deterrent to nuclear war before a freeze is feasible.

The extent of the administration's concern was underscored by a flurry of calls to House members from Reagan's chief arms negotiator in Geneva, Gen. Edward L. Rowney.

"I didn't call him back," confessed Rep. John LeBoutillier (R-N.Y.), who had co-sponsored both the Zablocki and Broomfield resolutions. "It was too expensive."

LeBoutillier had promised his vote to supporters of the freeze but switched at the last minute. "In times of need, you help the party," he explained.

If the president thought the vote was important, so did a man who might want to be president.

Minutes after the vote--the outcome of which was far from certain--Sen. Edward M. Kennedy's (D-Mass.) staff produced a statement quoting him as saying that the freeze "will prevail at the polling places in November and beyond."

House Democrats, despite a heavy budget and tax agenda, whizzed the freeze measure through the Foreign Affairs Committee before the administration could propose a substitute, and brought it quickly to the floor with the support of Speaker Thomas P. (Tip) O'Neill Jr. (D-Mass.).