Willful deceit and endangering your spouse is not a "lifestyle choice", it's wrong in an objective ethical sense.
Exposing your spouse to sexually transmitted diseases without their knowledge harms society as a whole. In cases where it's a wife cheating on a husband, forcing men to pay to raise children who are not their own harms society as a whole.
I'm not in favor of this kind of vigilantism either, by the way. But I think the relativism you express above, implying that cheating is a personal decision without a victim, is desperately in need of a reality check.
If people really thought that were true, it wouldn't happen. Why go to all that effort unless you really believe in your cause? The most likely outcome is that given a list of millions of names, the average person will search for public figures and people they personally know, and not give a shit about the rest.
Outing a cheater which you have no relationship with is a disgusting violation of the privacy of all parties involved, including the partner who was cheated on. It is interesting how the anonymity of the internet suddenly makes people more comfortable with shaming other people whom they have no personal relationship with.
I guess I'm just opposed to having a society where people punish strangers for not living up to their own moral codes. If we accept that someone reveals the names of cheaters on the internet, then we are accepting a society where it is okay to meddle in the most intimate parts of other peoples lives, even if their actions do not affect ourselves or society as a whole. This is wrong and disgusting in the same way as it is wrong when vigilante "sharia patrols" roam muslim neighborhoods in London, to bring up an admittedly extreme example. It is none of my business how other people lead their lives, even if I disagree with their lifestyle, as long as their actions do not harm me or the society as a whole. There is a reason we do not have a karma police.