Sounds like you are trying to tell the manager what to do, even though he might not consider it critical enough to be worth doing.
I suspect that the manager does give you the data when it is mission critical. When it is not, he focuses on other things. This does not necessarily mean he is ignoring your request -- instead, "yes" may mean "I heard you and I will consider it."
If he gives it to you eventually then great. If not then you should probably assume that it is not critical enough and refocus on things that are, unless informed otherwise.
One thing I would advise not to do is trying to force the manager to change his communication style only because you find it frustrating. It should be a lot easier (and less risky) for you to change how you interpret what the manager tells you, and adjust your own expectations instead.
Do continue to document your requests. That way when you give the manager an analysis memo that has gaping holes because you could not include the data you asked for but he never gave you, if he asks about it, you can point to 2 emails you have sent: 1 request and 1 reminder, which should be sufficient evidence of reasonable effort on your part.
If you get to the point where you absolutely must know the reason why data xyz was not provided in spite of your repeated requests, there is nothing wrong with setting up a quick meeting with the manager and asking him straight up, "I was wondering if you could offer any insight why we haven't been able to obtain data xyz for this analysis. I imagine there are some constraints I may not be aware of but it would be helpful for me to know how much effort is involved in obtaining this kind of data, so that I might consider workarounds." Good luck!