One of the less-noted in Office of Management and Budget Director David Stockman's wishbook would result in a 20 percent budget reduction for the largest law firm in the country -- the Legal Services Corporation.

LSC, one of the last major survivors of the War on Poverty, often is the legal mouthpiece for the poor and near-poor served by other programs Stockman would eliminate in his war on government spending.

The inclusion of cuts in the LSC budget is a small wave in the immense sea of budget reductions -- $64.2 million out of $45 billion.

But the ripple effect is much broader. An intense, protracted struggle over how the supposedly politically independent agency will operate -- or whether it will exist at all -- is about to be fought in Washington.

From the moment the legal services program began in 1965 and throughout its growth to a current level of 5,500 lawyers and a budget of $321 million, it has been a favorite on many conservative hit lists.

The federally funded activist LSC lawyers not only serve individual low-income people but, to the dismay and anger of conservatives, also lobby Congress as well as state and local governments for liberal changes in the laws affecting the people they represent.

In one instance, legal services lawyers convinced the Supreme Court that the U.S. Department of Housing and Urban Development was overcharging residents in publicly funded low-income housing. As a result, HUD may have to refund as much as $250 million.

There have been constant amendments introduced on Capitol Hill to prohibit the program from taking cases involving certain types of clients: children, gays, welfare recipients, undocumented workers or other groups. In most instances, these amendments have failed.

Still, an amendment to bar legal services from representing children barely was defeated last year. Another amendment to bar all program lawyers from representing clients before legislative bodies lost in the Senate by only three votes. With the new Republican-controlled Senate, supporters of the move say they are certain it would succeed now.

Legal services backers do not see a friend in the White House. As governor of California, President Reagan constantly feuded with the program in that state and once tried in vain to abolish it after it successfully stopped him from cutting Medicaid benefits. His attempt to veto funding for legal services was overruled by the federal Office of Economic Opportunity.

One part of Stockman's proposal is that legal services funding be part of a social services block grant which the states could divvy up among programs as they see fit. Had Reagan had that power when he was governor, abolishing the California program would have been easier.

If the program's backers defeat a move to bring it under control, and if all restrictive amendments fail, there is still the corporation's 11-member board of directors. President Carter failed to fill five openings before he left office; the other six become vacant in July. Reagan, therefore, has the opportunity to pick an entirely new board of his liking.

The new president already has some suggestions on the future of LSC. One transition report assumes the program can't be abolished and instead calls for shifting about half the agency's funding to a concept called "Judicare."

Under that approach, the government would pay private lawyers to represent the poor rather than providing salaries for full-time poverty lawyers. Without full-time staffs, opponents of the present system believe, there would be no lobbying and few class-action lawsuits.

It is not clear whether all or any of the recommendations of Stockman or the transition team will be proposed formally by Reagan in his budget message Wednesday, nor how Capitol Hill would react. No matter. hLegal Services Corporation backers say they already are running scared.