Can I quote from a viva examiner's report some positive feedback in a postdoc application? If yes, how do I cite the report?
(Expanding my comment into an answer, as suggested by Bryan Krause.)
I think it'd be unusual, and I honestly feel a little weird about it. Perhaps it can be done well, but consider how it would look before doing so. If you're using the quote in support of claims in your research statement, I'd wonder a) is there no support in the literature for that claim (why?), or b) if you're not confident in your own claims. If you instead hope to use the quote in support of your abilities or talent, I think it'd be better to ask the report writer to also write one of your recommendation letters. After all, if they wrote so in the viva report, chances are that they highly enough of you and your work to write a good letter. (It'd also look a lot better on paper than "X said I'm awesome!")
The second aspect to consider is if the quotes can be verified. In the systems I'm somewhat familiar with, these reports are usually not publicly available - at least not easily so. This means that those reading your application would likely be unable to confirm that the quotes are valid, making the value of including them kind of dubious. What's more, the report writers might have an expectation that the report isn't made public, and might disapprove of you using their words this way. (Now, you could of course solve that by asking them, but again, if it's central to your application, a recommendation letter would make the point more convincingly.)
Finally, to address your second question: If you choose to do this, you could probably just cite it as company report or internal report, ideally adding a link if it's available online. If your university do make these reports public in a more official manner, it might also be classified as a technical report, but I sort of doubt it. Generally, however, exactly how you cite something in your application probably matters a lot less than what you cite.