Mozilla, the Mountain View nonprofit that took on Microsoft’s Internet browser dominance nearly a decade ago and won, now wants to play the same transformative role with the mobile Web.

Mozilla is expected Wednesday to announce plans for its own app store, to be called the Mozilla Marketplace, offering mobile apps that could run equally well on an iPhone, an Android phone or a Windows Phone device. Mozilla is also working to develop a smartphone that would not be locked into the “walled gardens” of apps, operating systems and devices that are now controlled by Apple (AAPL), Google (GOOG), Microsoft, Amazon and a few others.

Mozilla’s vision is to develop a phone that would run apps within the phone’s browser, and that would not be limited to a specific operating system, such as Google’s Android, Apple’s iOS or Microsoft’s Windows Phone 7. The project is code-named “Boot to Gecko,” (a name based on a component of its Firefox browser) and Mozilla, which is talking to manufacturers and wireless carriers that might build and support a Mozilla phone, could unveil an early prototype of the software as soon as the Mobile World Congress in Barcelona, Spain, next week.

Mozilla hopes to re-create in the mobile arena the same sort of success it had with its Firefox browser, which challenged Microsoft and ultimately helped give consumers five strong choices for browsers.

Since the launch of Firefox in 2004, “our mission has not changed, it’s just that our world has moved on,” said Jay Sullivan, Mozilla’s vice president of products. “Instead of your choice for a particular (mobile) device narrowing your options, we’d like to see the user have control over their online life.”

Firefox now has about 450 million users, but access to the Internet is increasingly going mobile.

The problem, in Mozilla’s view, is that the mobile Web is increasingly dominated by a few powerful companies.

For example, if you buy an iPhone from Apple, you have to use Apple’s App Store and iTunes to download apps and media, and can’t access content and software from Google’s Android Market.

Given the massive and growing popularity of smartphones and tablets, it’s not clear that consumers are dissatisfied, or how much demand there will be for the alternatives Mozilla wants to offer.

Ken Dulaney, an analyst with Gartner who has been briefed on Mozilla’s plans, called them a win for software developers and “an interesting business challenge to the traditional lock-in ecosystem,” but said it’s hard to know how much demand there will be.

“We have a path over the hill. We don’t know if there’s a big city, or a little town, on the other side,” Dulaney said.

Mozilla is also trying to have an impact in the area of privacy and social networks. With Facebook, Google, Yahoo (YHOO) and Twitter all vying to become a kind of online passport system for hundreds of thousands of websites, Mozilla is building its own credential system, dubbed “BrowserID,” that would allow users to register and login to a website without sharing large amounts of their personal data.

“What’s different about ours, is that when you use those systems, a bunch of information-sharing happens behind the scenes,” Sullivan said. “So if you use a (Facebook) ‘Like’ button, or you even visit a page with a ‘Like’ button on the screen, some information is traveling behind the scenes.”

Mozilla won’t say whether it will unveil an actual “Boot to Gecko” phone in Barcelona, but its own internal Web documents called for developers to be using a prototype as their day-to-day phone by the last quarter of 2011.

Mozilla’s phone would run critical software like JavaScript and HTML5, the newest programming standard for the World Wide Web, directly on the device’s hardware, without the need for an intermediate operating system like Android or iOS.

In essence, the phone’s browser would become the operating system, supporting features like the camera, voice communication and geo-location abilities of the phone.

Mozilla is planning to open its app store to users later this year, while a commercial version of a Mozilla phone is probably significantly further away.

Still, Brendan Eich, chief technology officer for Mozilla, said the development of such a Web-powered phone is not some distant vision.

“Hang onto your hats; we’re moving fast,” Eich said. “A few years down the road, with Web technologies continuing to evolve, you should be able to do all the beautiful games and the presentation and the tools that you see” in native mobile apps, but with a Web-powered phone.

Contact Mike Swift at 408-271-3648. Follow him at Twitter.com/swiftstories, Facebook and view his Google+ profile.