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Some of the tension has eased in recent days. It has rained again, and more vaccines are on their way. Still, the confluence of crises is creating a rare opening for the opposition Kuomintang, or Nationalist Party, which has struggled for a path back to relevance and which favors closer ties with Beijing.

Ms. Tsai—who thumped the Kuomintang last year to win a second term in office—has seen her popularity plummet to below 50% for the first time since her re-election in one poll run by a former member of her party.

The crises have dented her image as a pragmatic and capable technocrat, and complicate her efforts to maintain a delicate status quo with an increasingly assertive Beijing, which never ruled the democratic island but claims it as part of Chinese territory.

Though Ms. Tsai is unable to run again for re-election, the crises are chipping away at the political fortunes of her Democratic Progressive Party.

“Popularity and elections are not our priority at this moment. It’s people’s health,” a spokeswoman for Taiwan’s Presidential Office said, adding that the administration is aware of and open to the criticisms.

With Covid, the Taiwanese leader is in part a victim of her own success. More than 2½ weeks of daily triple-digit increases have brought the island’s total number of cases to 10,956, with 224 deaths. Those numbers are relatively small but still startling for a population that previously had fewer than 1,200 cases, thanks to a swift response to the initial outbreak last year.

“The current outbreak surely has an impact on the government because people now have very high expectations,” said Ho Ming-sho, a sociology professor at National Taiwan University, noting the island’s success in keeping the pandemic at bay for much of the past year.

The island is badly lagging behind other developed Asian economies on vaccinations, with around 2.8% of its 24 million people having received their first shot as of June 4. That is partly because of the slow procurement, and the fact many Taiwanese didn’t feel an urgency to get vaccinated before the new surge. The new outbreak, traced to crew members on an inbound flight in late April, has fanned fears that Taiwan’s healthcare system could soon be overwhelmed.

On May 26, the head of Taiwan’s top medical institution, the National Taiwan University Hospital, posted a plea for more resources on his personal Facebook account, saying the hospital’s intensive-care-unit beds were already full. The following day, the Taipei Doctor’s Union warned that the island’s medical capabilities were being stretched to their maximum. “If this is not a breakdown of the healthcare system, then what is a breakdown?” the union wrote in a statement.

Beijing has said it is willing to provide Taiwan with vaccines, an offer the island’s health minister rejected, saying Taiwanese people wouldn’t dare use them.

Meanwhile, the Kuomintang has criticized the Tsai administration for requiring imported vaccines to have documentation showing they come directly from the factory, which it said discouraged companies and religious groups from donating vaccines procured on the open market.

“Why is the government still looking for excuses to refuse vaccines and finding thousands of reasons to impede multiple channels for vaccine acquisition?” KMT chairman Johnny Chiang said this past week.

The KMT said it is putting people’s interests first. Fan Chou, a political commentator and author of multiple books on Taiwan-China relations, said the party appeared to be “using the pandemic to win votes in future elections.”

Mr. Chiang said it is the opposition party’s responsibility to keep the government accountable.

“As a political party, our attention to saving people’s lives obviously takes priority over political considerations,” he said in written comments sent to The Wall Street Journal.

The Tsai administration spokeswoman rejected the KMT’s criticism on vaccines.

A plane carrying 1.24 million vaccine doses from Japan landed in Taipei on Friday. Taiwanese Health Minister Chen Shih-chung said earlier in the week that a plan to administer one million shots a week would kick off once 20 million vaccines the island has procured begin to arrive in late June. The Taiwanese government has also preordered 10 million domestically made vaccines, which Ms. Tsai said could be available as soon as July.

Meanwhile, the outbreak continues to spread despite social-distancing measures. Daily tallies of newly reported cases recently climbed back above 400 after dipping into the 300s earlier in the week.

Pressure from the pandemic comes on top of other tests. Taiwan’s worst drought in half a century has hobbled the island’s semiconductor industry, a major engine of economic growth, and contributed to large-scale blackouts in Taipei and other major cities.

While recent bouts of rain have eased the drought—and on Friday, a rainstorm turned streets in downtown Taipei into rivers—blackouts continue to affect parts of the island.

The blackouts illustrate how the government has overlooked risks in the design of Taiwan’s electricity system, said Hung Sun-han, an environmentalist-turned-lawmaker for the ruling Democratic Progressive Party. Taiwan typically relies on hydropower as a “relief pitcher” during times of high electricity demand, he said.

State-owned Taiwan Power Co. blamed the blackouts on human error and maintenance schedules.

“The DPP has been in power more than five years—enough time to fix things—but it hasn’t done the work,” said National Taiwan University’s Mr. Ho, citing the persistence of long-running management issues at state-run enterprises.

Another example, he said, was the island’s train operator, Taiwan Railways Administration, which has been held accountable for the deadliest train crash in decades that caused 50 casualties in April, following a similar derailment in 2018 that killed 18 passengers and injured 187.

A spokeswoman for Ms. Tsai said that “state-owned enterprises need to be reformed and are undergoing reform…People will eventually see the changes that are gradually taking place.”

Though Ms. Tsai faces no re-election pressure in her second term, her popularity will play a role in the August referendum, which includes a vote on pork imports that could complicate trade-deal negotiations with the U.S. She has portrayed the vote as important for pushing back against economic coercion from China.

“As long as the problems remain unsolved, public satisfaction with the DPP will continue to fall,” said Ting Jen-fang, a political-science professor at Taiwan’s National Cheng Kung University.