The Israeli Army moved large numbers of reinforcements into southern Lebanon today as it searched for two Israeli soldiers captured by Moslem gunmen yesterday and as fighting broke out between search units and Lebanese militiamen.

A 19-year-old corporal on an Israeli gunboat supporting the military operation from off the Lebanese coast just south of Tyre was killed when Moslem gunmen opened fire on the vessel. He was the third Israeli serviceman killed since Israel withdrew most of its troops from Lebanon last June.

As the Army's mechanized infantry and helicopter-borne troops moved well north of the Israeli-declared "security zone" in southern Lebanon and into areas patrolled by U.N. peace-keeping forces, Prime Minister Shimon Peres said Israel will not rest in its efforts to find the two Israeli soldiers captured during an ambush yesterday.

"For us, every person is an entire world, and we will not let up until we exhaust all the possibilities of discovering their whereabouts and seizing the guilty," Peres said.

However, despite extensive house-to-house searching in villages near Kounine, where the ambush took place, and well north of the security zone, no sign of the missing Israelis has been found and it was suspected here that they were taken from the area, either to Beirut or to the Syrian-controlled Bekaa Valley.

The Islamic Resistance Front, an umbrella group of radical Shiite Moslems, has claimed that it is holding the two Israelis.

The front, which is backed by Syria and Iran, said in Beirut today that the captives have been "moved to a safe place" for treatment of wounds, Washington Post special correspondent Nora Boustany reported from Beirut. Last night Lebanese television played footage supplied by the front that showed a man, said to be one of the captives, with his head bandaged.

The front said late today it would kill one of the two soldiers unless Israel pulled its forces out of Lebanon within 24 hours, Reuter reported from Beirut.

Another Moslem extremist group said today that in retaliation for the Israeli raids in southern Lebanon, it had killed a Lebanese Jew, kidnaped a year ago, The Associated Press reported.

[The Organization of the Oppressed on Earth, believed to be a Shiite faction, said in Beirut that it killed Elie Hallak, 52, a physician and vice president of the Higher Council of Lebanon's Jews, who was seized Feb. 21, 1985, in Moslem west Beirut.]

Security sources in southern Lebanon said hundreds of Israeli Army vehicles, including tanks, crossed the border to engage in the search. They said the entire male populations of several villages were rounded up for interrogation and that a dozen Lebanese were arrested and taken away from one village.

An Army command spokesman said he was not aware of any fighting except for the attack on the gunboat near Tyre. But officials at the headquarters of the U.N. peace-keeping force in Naqura, Lebanon, said eight persons with gunshot wounds were admitted to a hospital in Tyre as a result of clashes with the Israeli forces.

State-run Beirut radio said the Israeli push was meeting with stiff resistance from Moslem guerrillas.

The Army said two Lebanese guerrillas allegedly involved in yesterday's ambush have been captured and are being held for interrogation, and that a large quantity of arms has been seized.

After last June's withdrawal, Israel exchanged 6,550 Palestinian and Shiite Moslem guerrillas for 12 Israeli soldiers. That exchange, which included Palestinians convicted of terrorist murders, aroused a storm of controversy here.