Usually in these stories, Batman or whoever leaves behind enough evidence to support a successful prosecution, along with the tied-up bag guy.

The vigilante broke the law to gain evidence, so all the evidence the vigilante obtained would be thrown out,

That’s actually an interesting situation. The fourth and fifth amendments put restrictions on the government, not private vigilantes. So if the cops just happen to find evidence in plain view, there won’t be a direct constitutional reason to suppress it.

Now if the local prosecutor has a pattern or practice of deliberately turning a blind eye to the unlicensed private investigators that routinely supply them with illegally obtained information, there’s probably a claim there. But it’s a lot more complicated to make that case than a straight-up 4th amendment case.