Depends exactly what he did to get his evidence, but citizens hacking into private accounts is not better than the state doing it. This guy is kind of a Batman, doing ultimately the right thing, but you don't have to like the way he do it.
So, would you prosecute Batman?
I don't think the comparison holds up completely. Batman punishes criminals. This guy only extracted evidence for a case. If you want to interpret the principle as strictly as possible, yeah it's the same. They're both doing things they don't have the authority to do. Extreme differences in order of magnitude and types of action carried out, though. I hesitate in even calling the guy a vigilante, because that's commonly perceived as someone (like Batman) who punishes people based on their personal conviction.
But to more directly answer the question, I would not support prosecution of Batman if I lived in Gotham City, where he's basically the only reason the city hasn't been wiped out. I don't know how it is in the comics, but in the movies, the police are supposed to be horribly corrupt when he shows up and just letting crime run rampant.
If I lived in a place that was relatively safe and had a police force and government with a lot of integrity, I would support prosecuting Batman.
And personally, I find the current style of government surveillance to be verging on vigilante behavior itself, and they're not doing it for the sake of true justice. This sort of makes vigilante behavior in response excusable, in my opinion, both in self-defense and to make up for the failings of the establishment. I wouldn't go as far as all-out vigilante justice, but if a police department doesn't even bother to gather evidence for a case where someone has been severely victimized... sure... get that evidence yourself, so long as you're not doing anything extreme to get it.