HOUSTON—Not even two weeks ago, this city had one of the weakest apartment markets in the country with tens of thousands of vacant units. Now landlords say prospective renters are lining up outside their doors and some have almost no units left.

Before the storm, Atlanta-based Gables Residential, which owns about 3,500 units in the Houston area, had a couple of hundred vacant units. Gables received requests for 2,500 units from 50 different companies, including energy firms, looking for furnished apartments for local workers and insurance adjusters who may need to bring workers in. It suspended its online leasing to avoid renting out units that had already been booked online.

“It is going to fundamentally change the housing market in Texas for a while,” said Cris Sullivan, Gables’ chief operating officer.

Hurricane Harvey, which dropped record amounts of rain in parts of Houston, poses unique housing challenges because it affected a much larger swath of Houston’s population than has been impacted by previous storms. The White House said 100,000 homes in total were affected in Texas and Louisiana. In Houston, neighborhoods flooded that have never flooded before, places where many people don’t have flood insurance, which is likely to make the recovery longer.

Finding housing for victims displaced by the disaster will be a major challenge for this metro area of 6.7 million. In Houston, many of the units that were vacant before the storm were in luxury high-rise buildings, a legacy of the postrecession oil-and-gas boom when Houston’s economy seemed unstoppable. Those units were largely spared by Harvey while affordable housing units suffered more damage.

Ty Counts, a rental agent in Houston since the 1990s, said lower-income residents in the city already had challenges finding housing, given that the majority of new construction has been at the high end. Houston’s biggest block of available rental housing lies within the urban core, with just over 2,200 vacant apartments found in the downtown area, according to RealPage, a real estate software and data analytics firm based in Dallas. Monthly rent for available units in that area averages nearly $1,800, compared with just over $1,000 for the city as a whole.

“For the people who have means and money and insurance, they’re going to be fine,” Mr. Counts said. He said he worries lower income renters will “really suffer because if all the cheap apartments are gone, where are they going to go?”

Jesley Romero and her husband and two young sons are staying with family temporarily, after being rescued Friday by ladder and boat from the house they had been renting near the Addicks Reservoir in northwest Houston. She said she wants to find an apartment in the same elementary school district but knows of nothing in the family’s price range of $800 to $900 a month, particularly since so many of the properties they might have afforded also flooded.

“We’re living basically in my brother’s living room, staying on a pullout couch,” she said, fighting tears. “Only time can tell what the future holds.”

The storm appears poised to transform a rental market from one where supply exceeded demand to the other way around, both for high-end and low-end properties. Renters will likely have to compete for a dwindling number of vacant units, and in some cases pay higher rents than they would have before the disaster.

Apartment researchers said they anticipate that rents could rise by as much as 10% over the next several months. Until recently, Houston rents were essentially flat. Many landlords also have been offering tenants who move in two or even three months’ rent free. Those breaks are likely to disappear.

Houston had roughly 70,000 available units before the storm, or roughly 10% of the stock in the city, according to Bruce McClenny, president of Apartmentdata.com, a Houston-based apartment data firm. Mr. McClenny estimates that more than 60,000 units may have suffered damage during the storm based on the city’s experience in previous hurricanes, meaning virtually all the empty units could be absorbed by renters who are displaced.

One of the largest apartment companies in the area, Greystar, which manages some 45,000 units in areas impacted by the storm, said about 2,000 of the company’s units sustained water damage.

“Demand is going to overwhelm supply immediately,” Mr. McClenny said.

After Hurricane Katrina hit New Orleans the apartment vacancy rate plummeted to 2.7% from 6.7% and rents rose by more than 30%, according to an analysis by Reis Inc. The impact on Houston is likely to be less dramatic because it is a larger market with more high rises that are likely to have suffered less damage.

Much remains unknown about how the housing situation is likely to unfold. Many landlords say their buildings have fared better than expected, with only first-floor units in some buildings sustaining significant damage. Others say they haven’t even been able to get to their buildings to assess the damage because roads are impassable or employees are dealing with their own damaged homes.

Some homeowners will be unable to afford both a mortgage and rent and may move in with family or try to live in their damaged homes. Others may flee the city altogether, or be forced out to suburbs more than an hour from downtown.

Most major landlords and managers in the area said they would freeze rents in the short term. Gables’ Ms. Sullivan said short-term or furnished accommodations, which many Harvey victims are requesting, always command premium pricing but otherwise her firm doesn’t plan to raise rents.

“It will be a tight, tight rental market. Inevitably there will be landlords that will take advantage of that situation and raise rents,” said Nick Barber, owner of UMoveFree.com, a company that assists tenants with finding apartments.

Neonatal intensive care nurse Brittnay Burrell had been at Children’s Memorial Hermann Hospital for five days straight when she came home last week to find her ground-level apartment flooded. The refrigerator was toppled over. Everything was ruined—her clothing, furniture and prized possessions, including her vinyl record collection.

She said she got another shock when she learned that there was a run on apartments in the midtown area near the hospital. She toured apartment buildings Friday and immediately filled out an application for a $1,270-a-month apartment, in hopes of landing one of the few remaining one-bedroom units.

“It’s pretty overwhelming, honestly,” Ms. Burrell said. “I’m just trying to keep my chin up.”

Write to Laura Kusisto at laura.kusisto@wsj.com and Valerie Bauerlein at valerie.bauerlein@wsj.com