Five years after hedge-fund billionaire Edward S. Lampert brashly merged Sears and Kmart, the storied merchants keep shrinking.

So on the fourth floor of its grand Chicago flagship that epitomizes the 20th-century approach to shopping, a team of 180 e-commerce whizzes is searching for fresh ways to sell Kenmore appliances and Craftsman tools in an age of iPhone apps and Twitter.

The group — a brain trust that includes veterans from Web stalwarts such as Amazon and Orbitz — is giving a digital makeover to Sears, the 124-year old merchant that rose to prominence on the strength of its eclectic mail-order catalog.

Over the past 12 months, Sears Holdings Corp. has launched a flurry of Web sites and mobile-phone applications in an attempt to stretch sales beyond the physical borders of its aging stores. The strategy, dubbed “Shop Your Way,” is to market millions of items virtually, as well as in retail outlets, while offering various delivery and pick-up options. As the main Sears Web site now boasts: “Buy online, then go to the store and get your item within five minutes.”

Among other things, Sears is testing a concept called MyGofer that consists of converting an entire Kmart into a pickup-only site for Internet orders, replete with a drive-through. Over the holidays, Sears managed to boost revenues with a new take on an old sales lure: online layaway.

Like most companies trying to exploit the Internet, the once-mighty Sears isn’t exactly treading new territory. What it is attempting is a fast game of catch-up to stay relevant in the brutal retail sphere. “We have taken the good ideas out there and evolved them,” says Imran Jooma, the company’s senior vice president of e-commerce. “We are trying to create an assortment online that is almost limitless.”

When Lampert combined Sears and Kmart into one company, he promised that the sum of the new entity would eventually be worth more than its parts. That hasn’t happened, as mass merchants such as Target Corp. and Wal-Mart Stores Inc. have continued to wrest away market share. Sears’s revenues fell 7.8 percent last year to $46.8 billion; its same-store sales have dropped off every year since the merger.

Management, meanwhile, has been in flux. Lampert — who holds the title of chairman and controls the majority of Sears’s shares through his ESL Investments Inc. and its related funds — has no chief executive in place. Kmart veteran W. Bruce Johnson has served as interim CEO since Aylwin Lewis was forced out in 2008. Lampert, who doesn’t like to fly, rides herd over most major decisions from his offices in Connecticut, according to current and former executives.

Jim Barr, hired away from Microsoft Corp. in 2008 for the job of online president, left the company late last year. Barr declined to discuss his departure, as did the company. Lampert and Johnson both declined to be interviewed.

In light of the turmoil, Sears’s 2.0 venture is now among the most closely watched in the industry. The outcome, say analysts, could also be a matter of life or death for the Hoffman Estates, Ill., company.

“When I walk through Sears, I feel I have to hold my nose sometimes. But if I know they have the lowest price because I researched it online, I will go, and others will too,” says Love Goel, the head of retail private-equity firm GVG Capital Group. “The question is whether there is time to grow the online business while the knife on the stores is falling.”

Sears executives say that the online business has increased by double digits in the past two years. billion in 2008, about 6 percent of total company sales — more than Wal-Mart’s $1.7 billion, according to trade publication Internet Retailer. During the critical 2009 shopping month of November, unique visitors to Sears.com and Kmart.com grew by 22 percent and 42 percent respectively, according to Compete Inc.

Most traditional merchants, including Sears, first launched their e-commerce efforts about a decade ago. But many initially managed online and offline branches as separate channels competing for the same customers, including Kmart, which operated a side venture called www.Bluelight.com.

The emphasis has shifted under Lampert. Known for ruminating over every expense, he is loath to pour money into Sears’s crumbling stores, say people familiar with his thinking. He is far more bullish on Web ventures, with their small capital requirements and vast potential audiences. While e-commerce represents between 5 percent and 7 percent of U.S. retail today, many in the industry believe the figure might grow to be as high as 20 percent.

“If they can do this right, it may save the company,” says Maggie Gilliam, president of the retail consulting firm Gilliam & Co. “There is a sea change happening in retail right now and it is not clear what stores are going to look like in 10 years, so why spend money now? It may make sense for some companies, like Wal-Mart, as a defensive posture, but that is not the position Sears is in.”

On a recent morning at the flagship Sears in downtown Chicago, the youthful, caffeine-fueled team of e-commerce designers and engineers was abuzz. Among their latest projects: mobile applications that tap into smart-phone global-positioning systems and offer customers merchandise based on their location: Yankees sweat shirts in New York, Dodgers caps in Los Angeles.

To test their phone creations, the engineers descend into the department store with handsets and grab customers in the aisles. To obtain feedback on Web sites, workers take shoppers upstairs to a small laboratory that resembles a recording studio.

Inside, testers sit across a plate-glass window from the subjects and speak to them via microphone while a camera watches their eyes wander the computer screen. A recent quiz gauges shopper reaction to a new application that lets people check out shoes from every angle.

“We are not batting a thousand. There are failed apps,” says Tom Emmons, a former programmer for the Orbitz.com travel site who spearheads Sears’s mobile e-commerce team. “But we are being asked to take risks, and we’re taking them…If you look at the demographics of our shoppers and iPhone apps, there doesn’t appear to be a lot of overlap. But people use this stuff.”

To Emmons’s surprise, Sears customers are buying things like lawn tractors on their iPhones as well as electronics and clothes, suggesting a potential audience broader than foreseen.

Sears has also been tapping into the power of social-networking sites as a way to drive sales and win customer loyalty. With the help of industry veterans at a Chicago company called Viewpoints Network, in the spring it launched sites called MySears.com and MyKmart.com. Combined, the sites now have 400,000 registered users.

Members can sign into the sites with their Facebook accounts to ask questions about products and review them. Company employees monitor the conversations to stay abreast of complaints and customer-service problems.

“You can quantify the value of the insights customers bring,” says Viewpoints CEO Matt Moog. “If consumers are rating products very low consistently, that would trigger the merchandise person to talk to the manufacturer — and reduce returns.”