Rocket launchers, precision-guided missiles and billions of dollars’ worth of other advanced American weapons have given Ukraine a fighting chance against Russia ahead of a counteroffensive. But if even a few of the arms wind up on the black market instead of the battlefield, a Ukrainian lawmaker gloomily predicted, “we’re done.”

The lawmaker, Oleksandra Ustinova, a former anti-corruption activist who now monitors foreign arms transfers to Ukraine, does not believe there is widespread smuggling among the priciest and most sophisticated weapons donated by the United States over the last year.

“We’ve literally had people die because stuff was left behind, and they came back to get it, and were killed,” she said of Ukrainian troops’ efforts to make sure weapons were not stolen or lost.

But in Washington, against a looming government debt crisis and growing skepticism about financial support for Ukraine, an increasingly skeptical Congress is demanding tight accountability for “every weapon, every round of ammunition that we send to Ukraine,” as Representative Rob Wittman, Republican of Virginia, said last month.

By law, U.S. officials must monitor the use, transfer and security of American weapons and defense systems that are sold or otherwise given to foreign partners to make sure they are being deployed as intended. In December, for security reasons, the Biden administration largely shifted responsibility to Kyiv for monitoring the American weapons shipments at the front, despite Ukraine’s long history of corruption and arms smuggling.

Yet the sheer volume of arms delivered — including tens of thousands of shoulder-fired Javelin and Stinger missiles, portable launchers and rockets — creates a virtually insurmountable challenge to tracking each item, officials and experts caution.

All of which has heightened anxieties among Ukrainian officials responsible for ensuring weapons get to the battlefield.

“It’s impossible, honestly, to ask people to go through their stocks all the time,” said Ms. Ustinova, the chairwoman of a committee in Ukraine’s Parliament that monitors the transfer of weapons, in an interview in the streets of Warsaw last month, as she rushed to catch a train to Kyiv.

At the beginning of the war, she said, “it was just about survival, and people were just passing around Javelins” to repel a column of Russian armor that bore down on Kyiv early in the invasion. While those sorts of weapons are now routinely tracked, it’s still “very difficult” to account for small arms, like rifles, or the millions of artillery shells that the United States and its allies have sent.

The scrutiny is heightened for Javelins, Stingers and other kinds of missiles, as well as small-diameter bombs, certain types of drones, night-vision goggles and other systems being supplied to Ukraine.

But Ms. Ustinova says she has seen zero evidence” of illicit arms transfers of the sort that would destroy Ukraine’s credibility and threaten at least a cutback in U.S. support.

“Once there is smuggling or misuse of weapons, we’re done,” she said.

So far, American officials said, there have been only a handful of cases of suspected arms trafficking or other illicit military transfers of advanced weapons sent to foreign conflicts that must be most closely tracked.

Currently, federal investigators are looking into reports of Javelin shoulder-fired rockets and Switchblade drones being sold online after being taken from Ukraine, according to an American official, who spoke on condition of anonymity to discuss a highly sensitive issue.

There was one confirmed report of a Swedish-made, anti-tank grenade launcher being smuggled out of Ukraine. But the theft was discovered only after the weapon exploded in the trunk of a car about 10 miles outside Moscow, injuring a retired Russian military officer who had just returned from eastern Ukraine.

Inspectors at the Pentagon, State Department and the U.S. Agency for International Development reported in March that they had “not yet substantiated significant waste, fraud or abuse” of American support that has been sent to Ukraine out of 189 complaints they received alleging misconduct.

A rare visit by American inspectors to a Ukrainian military facility in Odesa on April 26 found “no irregularities,” said Capt. William Speaks, a spokesman for U.S. forces in Europe.

The commander of NATO troops in Europe, Gen. Christopher G. Cavoli, told Congress late last month that he could recall only one case of attempted smuggling — some automatic rifles — since the war began. He said he remained “highly confident” in Ukraine’s ability to secure the nearly $37 billion in U.S. weapons and other security assistance that has been committed so far.

But the threat remains. In intense conflicts like the one in Ukraine, weapons are being used almost as quickly as they are received. That makes hand-held missile systems and other portable arms “vastly more difficult” to track, said Nikolai Sokov, a senior expert at the Vienna Center for Disarmament and Non-Proliferation in Austria.

Accounting for ammunition is “next to impossible,” Mr. Sokov said. He cited unconfirmed reports of Stinger missiles “roaming Ukraine free,” and said officials appeared to be trying to persuade Ukrainian citizens to return light arms they received to defend themselves last year.

“This is what happens in every large-scale, lengthy conflict, and I do not see any reason to think it may be different with Ukraine,” Mr. Sokov said.

In interviews and congressional testimony, more than a half-dozen American and Ukrainian officials described an assiduous but fallible process to track U.S.-delivered weapons.

Before they cross into Ukraine, arms shipments stop at military staging centers in Europe, where the weapons’ serial numbers are recorded into multiple databases that are viewed by American and Ukrainian officials. The serial numbers are rechecked along the delivery route into Ukraine to make sure none are missing. They are also used to identify weapons that have been lost and later reclaimed; arms that turn up far from Ukraine would indicate they were smuggled.

Ukrainian officials “track it as it goes forward,” General Cavoli told Mr. Wittman in the House hearing. “We watch over their shoulder.”

This past December, American officials began giving Ukrainian troops hand-held bar code scanners to instantly transmit the serial numbers of advanced weapons into an American database. The new process was part of the decision by the Biden administration to give Ukraine more authority to self-report how it is securing arms.

American military officials said the shift was necessary, given that fighting has largely prevented U.S. inspectors from visiting battlefield units. But American officials responsible for the oversight remain concerned they cannot personally confirm the weapons’ whereabouts.

At least some Ukrainian frontline units under constant Russian fire are still waiting to receive hand-held scanners, Ms. Ustinova said. Such battlefield assessments have been infrequent in other war zones, American officials said, as smuggling generally becomes a concern when entire containers of sensitive missiles or rocket systems go missing — not individual light weapons.

Ms. Ustinova said Ukrainian officials and troops were all too aware of the stiff criminal penalties not just for smuggling American weapons but also failing to report any losses — arms destroyed or captured on the battlefield. Each lost weapon system is investigated and its serial number reported to the U.S. Embassy in Kyiv, she said, “so in case it shows up, in Iran or somewhere, we’re not being accused of that.”

She said the 16-person committee she chairs has doggedly investigated news reports of Western arms meant for Ukraine that have supposedly turned up with gangs, terror groups and other criminals. But Ms. Ustinova said she has found no evidence those reports are true, and echoed American assertions attributing them to Russian disinformation campaigns to sow doubt about NATO support for the war.

Yet the scrutiny is wearing on Ukrainian officials, who are balancing their dire need for weapons against onerous expectations for tracking them.

President Volodymyr Zelensky of Ukraine revealed “a twinge of frustration” and an air of “How many times do I have to tell you?” when the issue was raised last month by a U.S. delegation to Kyiv, said Senator Lisa Murkowski, Republican of Alaska, who was on the trip.

But Mr. Zelensky agreed it is necessary, she said, to ensure the continued provision of American weapons and other security assistance.

“All it will take is a situation where we find that somebody, somewhere down the chain, has gotten a piece of military equipment and has sold it for personal enrichment, or misappropriated it in some way,” Ms. Murkowski said. “Because then it just gets that much harder.”

Thomas Gibbons-Neff contributed reporting.