One way to do this is:

BEGIN {
    unless ( eval 'use DesiredModule; 1' ) {
        require AlternateModule;
        AlternateModule->import();
    }
}
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However, I've also made the argument that using two different modules is not necessarily the best approach. There's ways to get modules onto systems where you don't have permissions (local::lib or even App::FatPacker), plus if you're going to be writing your own module anyway, why not just always use that?

Updates: Added the ->import call, as per choroba's post. It's often not required for OO modules. If you need to pass a specific import list, you can do eval 'use DesiredModule qw/some_func other_func/; 1' and AlternateModule->import(qw/some_func other_func/);. Another possible variation is that you can replace the inner block require ... ->import(); with eval 'use AlternateModule; 1' or die $@||'unknown error';. Stringy eval is often the wrong choice, but in this case it's fine because it's using a fixed string of code.