I'd expect the boundary condition to be whether those web apps were all exactly the same technology, or if there's evidence you're quick to learn and pick up new stuff. If you're like "I did this first thing in PHP but for the second, bigger project I moved to Python/Django with some client-side JS", your odds should be great.
Hiring highly skilled programmers that know exactly the technologies they need is what many companies aim for when hiring, and never accomplish. They're wrong. Even Google has problems finding (and hiring) those. That's why I said it was a mistake.
So, I would expect that you can find these jobs. And the companies that are willing to hire you might also be the ones you want to work at. But some HR departments will turn you down, and some people (like the original poster) may turn you down because it's a risk. (I can't blame them - hiring the wrong person is a bad mistake, too - but they shouldn't complain it's hard to find good people.)
Hiring highly skilled programmers that know exactly the technologies they need is what many companies aim for when hiring, and never accomplish. They're wrong. Even Google has problems finding (and hiring) those. That's why I said it was a mistake.
So, I would expect that you can find these jobs. And the companies that are willing to hire you might also be the ones you want to work at. But some HR departments will turn you down, and some people (like the original poster) may turn you down because it's a risk. (I can't blame them - hiring the wrong person is a bad mistake, too - but they shouldn't complain it's hard to find good people.)