Translation:We have no proof that they have stolen anything.
Simply bad English. How about "We do not have any proof that they have stolen anything"? The way that Duolingo is going here, why not do a full Monty and put "We ain't no proof that nobody stole nothing" as the suggested solution? Bonus points if you can picture Robert de Niro saying that.
to be totally, bluntly honest: I was told it was bad English in school ;-) There is a good chance this is either outdated (I am oooold :-D ) or I am just perpetuating my teacher's misconceptions. I googled, and didn't come up with anything ("and came up with nothing"?). What I am seeing as a logical problem here is, that the construction with any, aka "we do not have any.." properly refers to that you are actually not having it, whereas "we have no..." asserts first that you actually do have something, and then you negate that something, leading me to ask "when you do not have it, why do you start with that you do have it?". Might be that. But I am clutching at straws here and I am not entrenched enough to defend that position. It still sounds strange to me with "no", but the operative words here might as well be "to me".