Though this question has an accepted answer, I don't think it entirely answers the question as titled. We have covered why Bradley doesn't kill them but that's not the main question. Why did he allow them to continue their investigation? He didn't need to kill them to stop them from investigating. So why didn't he pressure them, took them captive or do similar things?
Well, the answer is: He did. He warned them about the danger these investigations pose to them, we see him dropping by several times, casually chatting with Winry, he meets Mustang with the Officers, ... Almost every time we see him, he at least implicitly threatens our protagonists. As Mustang continues his investigation, he becomes more direct and has Mustangs people transferred and Hawkeye as some kind of hostage. He even imprisons doctor Marco.
Keep in mind that we are talking about Wrath, who would like to kill every one of them just for the sake of doing so. He tempers himself at all time so it is hard for him to decide what the right amount of action/force is. He starts off with warnings and puts them under surveillance so he could have them killed when they found out too much (RIP Hughes), then he threatens them and reduces their options. He can't kill most of them so he does everything he can (except incarcerate them, but that would be super inconvenient for the plot) to make sure they wouldn't find out too much.
His only other option would have been a big bluff doubling downing on the story that he was trying to uncover the conspiracy and urging them to either work directly under him (have them under control) or to drop the investigation because it is too dangerous since it reached up to right under himself.
[Edit:] Another reason to neither kill nor incarcerate them is the fact that he still is a military leader in a time of open conflict with several neighbors. As hinted at in the series, after Scar killed quite some State Alchemists, the strength of the military is considerably weakened. Losing three more state alchemists and a living armor that would fight at Ed's side anyway does not appear to be a wise decision. Father doesn't want to waste his resources to defend the country so Bradley has to maintain the military's strength. And purposefully weakening the military at a time like this would appear odd to anyone else even if no one else could pose a thread, more people investigating would be an inconveniance.
[Edit2:] From Episode 30 on, neither Mustang nor the Elric Brothes conduct any investigation (or act against Father in any other way) that is visible to Bradley. After thier next visible steps, they become outcasts and deserters.