With how heavily indoctrinated the Spartan-II's were, it's highly, highly unlikely that they would go rooting around through classified ONI property. Spartan-II's are to protect humanity, with a secondary mandate to enforce/secure UEG control, no questions asked. Whatever Chief and Blue Team are doing, it is done with the goal of human advancement or security at its core. Now with Locke, we have the same goals. However, he is 100% ONI field operative, so his view of events are likely altered to make ONI look good. With Chief being declared rogue yet again, I think Locke has been tasked by ONI to hunt Chief down, but ONI are withholding some details. If Chief's loyalties lie with humanity, but not with ONI, there's not much ONI can do to change that. So why would they want him back? Because Chief is a figurehead. Or at least, the armor is. The individual inside the armor isn't necessary, and can easily be replaced with someone else. Just add a voice modifier and no one will know the difference. You secretly eliminate a rogue asset, publicly "return" the "asset" to loyal service, and the public opinion skyrockets. It's a win-win for ONI.

It's been heavily stressed by 343i that Locke isn't just an ONI goon. He never really was to begin with.