Science normally addresses natural events in terms of natural causes. Whoever agrees on that (at least approximate) definition of science thereby grants that science doesn’t prove or disprove God. It just doesn’t address God’s existence. It’s not the right tool for that job.
Science can, however, address effects in nature which might have supernatural causes. In this way science could discredit or lend credit to different lines of evidence for God. Remember, “science” is usually a reference to “natural science,” and so it addresses natural events and is usually restricted to natural causes.
If we are to connect scientific data to the question of God’s existence we need a mediator, like theology, philosophy, or even art. By that understanding, Science isn’t the right tool for directly addressing God’s existence.
The data we glean from science has to be passed off to other fields to consider how it might best be interpreted if natural events/causes seem unable to account for that data. Science can identify and clarify things that are not explained or cannot be explained by natural causes.
Science can also discredit/falsify some supernatural claims by pointing out sufficient but natural cause for an event. For example, debunking magic tricks and various fraudulent religious claims like spoon bending, levitation, mind reading, etc.
The question at issue here is, not about whether science disproves God, but is more subtle: “Does science preclude God?” This is a more sophisticated question because it allows that God may exist, but to do science we have to assume/believe/conclude he doesn’t exist. Doing science makes God-belief impossible, or, at least, does it make God-belief impossible while one is conducting science?
At minimum, this is a question about methodological naturalism–must we deal strictly in natural causes, allowing only natural explanations into the mix whenever we do science? I myself am torn on that question, depending on how one understands “methodological naturalism.” At maximum, this is a question about metaphysical naturalism–does the operation or outcomes of science somehow make God’s existence impossible? I’m not torn here.
THE SHORT ANSWER TO THIS QUESTION IS: NO, SCIENCE DOES NOT PRECLUDE GOD’S EXISTENCE. WE CAN CONDUCT SCIENCE AS WELL OR BETTER THAN OUR ATHEISTIC COUNTERPARTS PRECISELY BECAUSE WE ALLOW THAT GOD EXISTS.