Representatives of 13 American Protestant and Catholic religious groups sent a letter to congressmen this week lobbying against the MX missile, in what could become the biggest such campaign against a nuclear weapon since organized religion challenged the morality of dropping the first atomic bombs on Japan in 1945.

The effort is aimed first at the 55-member House Appropriations Committee, which is scheduled to vote later this week on an amendment sponsored by defense subcommittee chairman Joseph P. Addabbo (D-N.Y.) to delete money for MX production from the fiscal 1983 defense appropriations bill now before Congress.

"This week when you vote on the defense appropriation for fiscal 1983," the letter begins, "we strongly urge you to vote to delete the $989 million in the subcommittee's recommendation for procurement of MX missiles."

President Reagan has recommended to Congress that 100 MX missiles, each designed to carry 10 nuclear warheads, be placed in closely spaced silos outside Warren Air Force Base near Cheyenne, Wyo.

"The MX is a dangerous, destabilizing, first-strike nuclear weapon," the letter argues. "Its deployment would signal our leaders' belief that we can fight and prevail in a nuclear war."

It also contends that spending $26 billion on MX missiles "would rob the poor and needy" of resources available to them and that deploying the missiles in the "dense pack" formation "would make a mockery" of the government's promise to abide by previously negotiated arms control agreements. It also would "send the wrong signal to the new Soviet leadership.

"We reject the notion that we are behind the Soviets in the arms race," the letter continues, "and the idea that the way to achieve significant arms reductions is first to rearm. Indeed, we reject the assumption that weapons of mass destruction have any moral justification whatsoever. Now is the time to stop MX once and for all."

Religious groups represented by signatories to the letter are: American Baptist Churches; Church of the Brethren; Episcopal Church; Jesuit Social Ministries; Mennonite Central Committee; National Assembly of Religious Women; National Council of Churches; Network, a Catholic Social Justice Lobby; Unitarian Universalist Assn. of Churches in North America; United Church of Christ; United Methodist Church (Board of Church and Society, and Women's Division) and United Presbyterian Church.

Veteran House staff members working both for and against the amendment to cut off MX production money said yesterday they could not recall when religious groups last had mounted such an intense campaign against a single weapon.

A Pentagon official last night said he hoped a way could be found to delay the crucial vote in the House committee, but acknowledged that this did not appear likely.

In a related development, the State Department dismissed the Soviet warning that it would launch its tactical nuclear missiles as soon as allied ones based in Europe were spotted as "part of the Soviet propaganda campaign" against NATO plans to modernize its theater nuclear forces by deploying Pershing and cruise missiles.