here's the thing, if you're viewing it on a website then it's in your RAM. if it's in your RAM, it can be moved to your hard drive. unless they want to encrypt your ram, thus slowing down the entire PC because it has to encrypt and decrypt everything, there's no way to prevent people from doing this.

also, desktop level screen captures don't care about the running program, browser or otherwise. they'll capture whatever's rendered to your monitor. unless they want to intercept what's rendered to your monitor, there's no way to prevent people from capturing this data and storing it.