It would help if we knew how many jobs you have had in total and their lengths, so that we can judge how serious one missing job is.
Having said that, you say that it is the most relevant to the one for which you are currently applying, so that probably tells us what we need to know.
Having never been a professional front-end developer (I do embedded software), I can’t say what the contractual norm is.
I suspect that if you are freelance, you may be able to write a clause into your contract saying that you can use the work for advertising purposes and the like, although I would expect the client to (try to) forbid you to sell it to other clients. I.e the client owns copyright and you have limited permission to display the work.
For permanent positions, it depends on location, which you did not state, but in most places all work is the property of the company which paid you to produce it.
That’s quite clear for me – I can't show the code to anyone outside of the company. It’s a little hazier for you, as the companies have made the work publicly accessible.
However, in your place, I would restrict myself to walking though the web-site in-situ and explaining your part in it. If I were interviewing and you showed up with a copy of the website on your laptop or a USBB stick, I would be concerned that if I gave you a job you might walk off with our intellectual property. Off course, I don’t know your industry norms, so take that with a pinch of the proverbial.
As @DarkCygnus said, you might want to read Applying for jobs, new company wants to see source code that I don't have access to any more.
In my line of work, we have no past code to show at an interview (apart, possibly, from some private stuff; maybe a GitHub repo), so we talk our way through the functionality of the project and our part of it.
And that’s what I recommend you do here. Since you say that the “missing” job is the most relevant to the position that you are applying for, the interviewer should have no problems following what you say, and being able to explain something verbally, rather than just point to something and claim it is much more persuasive at an interview. Good luck :-)