you almost sound like you think the thief did nothing wrong. Do you just feel like all theft should not be able to be stopped by citizens? Only police?
Unless you are capable of overpowering the thief, that leaves you needing a weapon.
In my opinion, I turn your argument against the thief. You are in a state where much of the population has firearms. If you are dumb enough to steal, you are taking the huge risk of personal injury(or worse in this case).
Don't steal unless you know all the risks associated, and yes, being shot is one.
Of course the thief did something wrong. Multiple things, really. She used a public website to sell a service that is illegal (morally I'm fine with this, but it's still a really stupid thing to do, not to mention illegal), she scammed some poor guy and she was attempting to steal $150, which, while not a princely sum, is also a fair amount for most people. She was in the wrong.
So she got shot in the neck.
That's a bit of an overreaction, no? Can you think of a single jurisdiction on Earth that would give the death penalty out for stealing $150? There probably are some, but I'm guessing it's somewhere none of us would voluntarily live. Her punishment for a bad scam operation is death. That's overkill, no pun intended. The guy still had her contact info, he could have gone to the police, reported it to Craigslist, or any number of other actions. He kills her?
Not to mention, let's examine what he did wrong. He solicited sex from someone over the internet in exchange for money. Again, I don't personally find that immoral, but it is illegal. The entire transaction started with his commission of an illegal act. If he hadn't ordered a prostitute, he never would have found himself trying to get his money back from one. If you're going to play the "broke the law" card, he's equally culpable for setting up their transaction to begin with.
Petty theft does not deserve vigilante justice from someone who was engaged in a criminal act himself.