Is the expression "done right" proper grammar?

According to M-W.com, right (adverb) can mean "in a suitable, proper, or desired manner", and gives as an example of that usage "knew he wasn't doing it right." So I'm gonna have to go with the "you're just crazy" option. :-)

It may sound wrong because it's a contraction - the full form is "this is a website that was done right". Now that's ugly passive voice, I think "someone did a great job on this website" sounds better.

In other words, it's wrong in the same way as "as seen on TV" is wrong :-)

According to the Wiktionary entry for "right", there is no meaning of "right" that is an adverb meaning "correctly" (however, there are other adverbial meanings of "right"). So, if you trust it as a source, the (pedantic, technical, grammatical) answer is no. However, "done right" is very commonly used in informal contexts in American English to mean "done correctly". I would not say or write "done right" in any formal context.