There's no universal way to do it. You could use the platform's package manager to query for known browser names; this would miss browsers that were not installed by the package manager, and it's distro-specific (rpm, apt, etc)
You could ask the desktop environment; first, of course, you'd have to figure out which one was in place, then query it in an environment-specific way.
Finally, you could search for known executable names -- mozilla, konqueror, etc. If I had to do this, I think I'd do it this way.
depending on exactly what the linux platform is, the package manager might actually be of considerable help. packages often can list what "features" they provide, and some package managers can be asked what packages provide a given "feature". for example, on my Mandrake Linux system, i can do this:
here, the "webclient" feature should be "provided" by every package that can serve as a "web browser" of sorts. but there are several things to note:
some of the items make little sense. "kdebase-common", for example. well, that's because the KDE desktop environment has a "file manager" that can double as a web browser... but unless you knew that about KDE, you'd be unlikely to know what that item was about.
some of the items aren't what you'd think of as a "web client". wget, for example, just downloads files to your hard disk, it doesn't try to display them at all. is that a "web client"? well, it speaks HTTP, so...
it missed my firefox, because that wasn't installed using the package manager.
i had to know that "webclient" was the right "feature" to use. not every distribution will necessarily use that name for it.
not every distribution will necessarily use a package manager that can do this at all!
so, for certain specific cases — for example, if you decide to formally support only one linux distribution — this might be of some use. but in the general case, when you need true platform independence, Ernest's advice is far better; in those cases, you'll just have to search for executable file names. or ask the user which browser to start.