First, I agree with the claim stated in the currently most upvoted answer that the natural sciences do not prove things in the way that the formal sciences, like logic, math, and computer science, prove things, and that the natural sciences cannot give us 100% certainty.
However, as an answer to the OP I think the answer falls a bit short.
As I see it, we can easily take the use of the word 'proof' in the question to mean the word 'proof' as when in court we 'prove' that someone is guilty beyond reasonable doubt.
Indeed, I would say that scientists use the phrase 'scientific proof' in exactly that way, i.e. as 'demonstrated beyond reasonable doubt' rather than '100% certain'.
So, interpreted that way, I think the OP question still stands and is a very reasonable question: does it make sense to use science to try and 'prove' the existence of God in that sense of the word? Indeed, can we use science to obtain any kind of evidence at all?
Well, as others have pointed out, much depends on how one defines God. Certainly for some concepts of God it seems that science just won't be able to find any evidence; I am thinking of the kind of God who created the world but otherwise just sits back and watches from a distance as we are steadily fouling up the planet God so kindly provided for us.
However, if the God we pick has causal effects on the observable world on a more day-to-day basis, then it seems we might be able to use science to convince us that there is really 'something akin to a God' out there.
For example, if, as the aforementioned answer imagines, God indeed comes in every Saturday at NIST HQ riding his light-beam, well, then we can use science to try and see if there is indeed some genuine object there, or if maybe the people observing this are suffering from some kind of mass hallucination.
Now, of course, we have to be very careful here: if we find that the cup of coffee pops out of nowhere, well, that's pretty weird, and demands an explanation ... but I am not sure if that should convince us that this was due to the 'God' we had in mind, as opposed to something else. Indeed, if being 'a really, really, really, good being (and pretty powerful, to boot)' is part of how we define God, then frankly, producing coffee doesn't really impress me much, but I would be a good bit more convinced if this being snaps their fingers and all cases of bone cancer in children would disappear (apologies to Stephen Fry).
So, depending on how you define God, I think that yes, science could potentially find evidence of such a God, and maybe even 'prove' its existence beyond reasonable doubt.
Finally, let's be clear that science should of course not be in the business of trying-to-prove-God, but rather (assuming we find this to be a worthwhile pursuit at all) it should be in the business of testing-to-see-if-there-is-a-God. Indeed, in that sense, the original sense of the word 'proof', which was much closer to 'test', may in fact be the best of them all in the context of this question.