Is it possible to calculate sha256 hashes in the browser using the user's video card, eg. by using WebGL or Flash?

Looks like this can be done (although in this case it's not SHA256). The following is an example of a JavaScript library that uses WebGL2 to calculate hash values on the client side, for the Curl hashing algoritm: https://github.com/iotaledger/curl.lib.js/

In this case it's used to do Proof of Work for an IOTA transaction (https://www.iota.org/get-started/what-is-iota). This basically comes down to brute-forcing random inputs into the same hash function until the result matches a certain output. Therefore the gained hashing speed by using WebGL is very relevant. I have used it, and it works!