Moves by Congress and court may lower the wall between church and state.

Supporters stood all night in a cold drizzle, singing songs and rallying for prayers in schools, an idea said to have the backing of 80 percent of Americans. Inside, a band of about 60 congressmen took the House floor for nonstop speeches favoring a school-prayer amendment to the Constitution.

The scene at the United States Capitol was only the latest sign of an intense lobbying effort in which the chief lobbyist is President Reagan, who has put the amendment at the top of his election-year legislative agenda.

But despite the thrust behind it, the movement to put organized prayer into public schools still faces problems in Congress. Even in the Senate, where majority leader Howard H. Baker Jr. (R) of Tennessee has given the proposal full backing, the amendment is not guaranteed the required two-thirds majority and could face a filibuster.

Perhaps more significant, the amendment's supporters are not yet agreed on the exact wording or effects of it. Three versions of the amendment exist, and while two have been voted out of the Senate Judiciary Committee, the panel took the unusual step of making no recommendation on either one.

Now before the Senate is language proposed by the Reagan administration which states:

''Nothing in this Constitution shall be construed to prohibit individual or group prayer in public schools or other public institutions. No person shall be required by the United States or by any State to participate in prayer. Neither the United States nor any State shall compose the words of any prayer to be said in public schools.''

(Addressing the National Association of Evangelicals in Columbus, Ohio, President Reagan affirmed his support for the bill Tuesday: ''I firmly believe that the loving God who has blessed our land and made us a good and caring people should never have been expelled from America's classrooms. And the country agrees.'')

Proponents are divided on who would select the prayer. Senator Baker says that the ''content of the prayer should be voluntary'' and that students themselves would give the oral devotion. He opposes allowing local school boards to authorize official prayers. Mr. Baker has held out the possibility that students be given a time period for prayer in which each could pray aloud as he chose.

But another prime supporter disagrees. ''I think there will be a school prayer chosen'' by the local school boards, Sen. Orrin G. Hatch (R) of Utah says. ''And once it's chosen, that'll be it.''

The committee report on the amendment holds out both possibilities. After passage of the amendment, ''it is likely that most prayers will be developed by the students themselves,'' the report says.

''Another alternative would be the selection of a prayer from religious literature,'' the judiciary report adds.

Gary Jarmin, director of the Project Prayer Coalition, says the final version will probably ban school boards from writing or selecting official prayers. The effect would be that ''the teacher would stand up and say, 'Would anybody like to offer prayer?' '' Mr. Jarmin says. ''If not, there would be just a moment of silence.''

Also uncertain is whether children who objected would be allowed to leave the classroom. While the committee report indicated that they would, Baker and others have said that a school board, under its discipline authority, could require all students to remain, even if not participating.

''If each senator has a different interpretation, we're not going to pass it, '' a GOP Senate leadership aide conceded Tuesday. During the next two weeks, he said, the ''legislative history will show what is intended.''

But two weeks is not enough time, says Sen. Lowell P. Weicker Jr. (R) of Connecticut, the leading opponent of the amendment. He gave notice this week that he wanted to delay the vote until June 1, although a two-month debate might be acceptable.

''Children can pray right now - when they go to the lunch hour, when they go to bat, before a math exam,'' Senator Weicker told reporters. ''They can pray on their own.

''I can't think of anything I would fight as hard over as freedom of religion ,'' said the Connecticut senator, who two years ago staged an 11-month fight on another constitutional issue, removal of federal court jurisdiction.

''I don't want my children to be told how to pray,'' said Weicker, who added that he believes that with time the public tide will turn against the amendment.

The basic issue is civil rights for children from religious minorities, says the Rev. Charles V. Bergstrom, director of the office for governmental affairs of the Lutheran Council. He and leaders from several mainline Protestant and Jewish groups have formed a loose-knit coalition to oppose the amendment.

''Prayer is such a meaningful, deep communion with God that schools should not be involved in any way in deciding what prayer should be used,'' says Mr. Bergstrom. Lutheran churches backed the 1962 Supreme Court ruling that forbade schools from organizing verbal prayer.