The political heirs of the late Eric Williams, Trinidad's iron-willed leader for 25 years, have overcome their first political challenge, but now must find solutions to the country's serious political problems.

George M. Chambers, 53, led Williams' People's National Movement to a stunning election victory in November, despite a challenge from a new party that attempted to woo the black voters who had traditionally made up Williams' political base. Under Chambers' leadership, the People's National Movement won its sixth election in a row with the largest popular mandate ever--219,000 votes and 26 out of 36 seats in the House of Representatives.

Williams, who died in office at the age of 69 last March, "has left intact the most effective political organization in the Caribbean," said the victorious Chambers in a recent interview.

The election was closely monitored by U.S. diplomats here, since the relatively wealthy, twin island nation of Trinidad and Tobago plays a prominent role in the Caribbean.

The Reagan administration has made the region one of its top foreign policy concerns, trying to stem what it describes as an ominous growth of Cuban and Soviet influence.

With billions of dollars in earnings from petroleum sales to the United States, Trinidad's major trade partner, this country of 1.2 million people in recent years has undergone a rapid and stressful modernization. But the oil revenues so far have failed to solve domestic problems--unreliable utilities, lack of an effective public transportation service and a potentially explosive absence of low- and moderate-income housing for a population of which 62 percent is under the age of 35.

The underlying conflict between mainly urban and Christian blacks and the more rapidly growing rural population of Hindu East Indians that was the key issue in the November election causes political instability that threatens to exacerbate Trinidad's economic problems.

An eruption of black-power demonstrations and widespread dissatisfaction with the government almost brought Williams down in 1970. Trinidad has exported petroleum since early this century, and the windfall revenues after the 1973 price increases helped reestablish stability and Williams' grip on power. But he grew increasingly eccentric and reclusive after those disturbances and only rarely was seen in public.

With the oil revenues, Trinidad's once agriculturally based economy grew by 10 percent annually and boosted per capita income from a few hundred dollars to $4,700 today. This year, with crude oil production of 190,000 barrels a day, Trinidad will earn about $4 billion. Trinidad has an enviable $3.1 billion in foreign reserves, enough for two years' imports.

But the oil boom has also brought problems: inflation runs at 18 percent annually. Five local plants assemble mostly Japanese autos and others are imported to put 2,000 new cars a month on already congested streets. This year's food import bill, for a country that fed itself two decades ago, is expected to climb above $350 million a year as the agricultural sector continues to decline.

Trinidad's oil reserves, at declining rates of production, are expected to last for at least another two decades, but its proven reserves of 10 trillion cubic feet of natural gas could form the basis of continued affluence well into the 21st century, economists say.

It was the natural gas that Williams had in mind when he invested $1.1 billion in a controversial industrial complex on Trinidad's southwest Point Lisas peninsula. The project has raised widespread criticism and charges of mismanagement and corruption, a major issue for critics of the People's National Movement.

An alliance of the mainly East Indian Labor Front and two small non-Indian parties has announced that it will take the initiative against alleged corruption, inefficiency and financial mismanagement in Chambers' government.

Williams' stature as father of Trinidadian independence, Caribbean historian and former university professor "allowed him to ignore" these charges in the past, said East Indian Labor Front leader Basdeo Panday. "We do not feel that Chambers has that ability."

The government's political critics also charge that a number of Williams' ministers have been involved in kickback schemes involving the purchase of airplanes for the national airline and such large prestige projects as the construction of a multimillion-dollar Hall of Justice building in downtown Port-of-Spain.

Williams had become so powerful, his critics say, that there was little parliamentary discussion about any of these projects or the corruption allegations.

"For example," said House opposition leader Panday, "there was a widespread feeling that the Point Lisas projects could become white elephants," but Williams was able to ignore any efforts to have the investment thoroughly debated.

In an hour-long interview in his White Hall office in downtown Port-of-Spain, Chambers, a former minister of agriculture, finance and industry and commerce, said the government would be reforming some areas of mismanagement and establishing new priorities where there are pressing needs.

Public transportation and moderate income housing construction would be two of his "top priorities," Chambers said. The decline of the country's agricultural sector would probably continue, Chambers indicated, as "you don't legislate to put people on the land."

One of the government's constant critics in the parliamentary opposition, John Humphrey, said in an interview that the Point Lisas steel plant, completed in 1980, was losing $1 million a day.

Independent figures were not available, but Chambers denied Humphrey's charge.

"No report has reached me on that," said Chambers. "I think it is unfortunate that while the country is trying to move into the technological era, people who should know better spread unsound reports and stories that are not in the national interest," he added. He also said he would appoint an official who would be "taking steps to correct misinformation" about all the Point Lisas industrial projects.

Chambers, who has a reputation of having been personally "clean" during a long government career, said some allegations about corruption "have been wild, some not so wild." He added: "I'm on record as stating that I intend to be quite ruthless in dealing with corruption."