This is how it always works. I was thinking, though, that if you, same as becoming a Muppeteer, sent them a tape of some of the puppets you've built, show them what you're capable of, something might come up, and they may think of you. That probably wouldn't work, but, it's worth a shot.
Generally in theatre/TV/film for puppet building job you either present the puppets in person or submit/show a portofolio with photographs. While there's no rule against showing videos to get puppet work (I know there are a few builders and companies that do this), portfolios are generally easier for whoever is hiring to review. Also, with video as a builder you run the risk of being judged on the basis of the performance rather than the work itself. IF you were to submit a video for a puppet building job, I'd suggest sending photos as well. Most companies will either state their submission guidelines when they're actively searching or have general guidelines for accepting unsolicted material.
I know some people have mentioned in the forum in the past that they'd like to get a job building and then eventually use that "in" to cross over to puppet building. Henson actually discourages this, saying there is very little crossover between the workshop and the performers (though Dave Golez started out in the workshop in the 70s).
Maybe Toastcrumbs could explain whether or not this is true...
A friend of mine was invited (several years ago) to be a guest puppeteer on Muppets Tonight. While she was hanging out with Steve Whitmire, he suggested that she gain performance experience by starting her own cable access TV show.
Many towns/cities offer FREE classes on how to produce your own TV show. I used to work in the cable TV industry, and trained MANY people. The studio I worked at had very nice equipment, and it was available to the public to use...FOR FREE! Not many people used the studio, so I would make music videos for local bands.
Now that I have got back into puppetry, I am scripting a series that I will produce and direct at our public access TV studio.
Check what your cable system offers. This would be one way to learn how to perform for TV, and make a demo reel to show to future employers.