Chirrut was not "Force-sensitive" - that is, he could not use or control the Force reliably. Given his age, he almost certainly would have been recruited by the Jedi if he had any noteworthy Force abilities, and would have been caught up in Order 66.
From the novelisation of Rogue One:
"Who are they?"
"The Guardians of the Whills. Protectors of the Temple of the Kyber. But there's nothing left to protect, so now they're just causing trouble for everyone."
The closest we get to a direct statement of his Force sensitivity comes a little later:
"Is he Jedi?" he asked, with the hushed doubt of a man on the verge of a great discovery.
"No Jedi any more. Only dreamers like this fool."
So if Chirrut isn't a Force user, how do we explain his preternatural dodging skills? Well, Catalyst made it clear that people who cannot use the Force can still feel the Force:
So I think we've got enough information to clearly answer the question.
Chirrut was not a Whill. He was certainly not a Shaman of the Whills. Chirrut was a member of a group called the Guardians of the Whills; this group guarded and protected the Temple of the Kyber on Jedha. Those who guard a group do not themselves have to be members of the group; Royal Guards aren't royalty, after all. While Chirrut could not control the Force himself, he did have the same slight sensitivity to it that all living things have. In his blindness, he learned how to use that sensitivity, and learned to trust it and trust his instincts.
He had no peculiar access to the Force, not the way Jedi and Sith do; but he has learned to make the most of the minimal access that he does have.
(I chose to delete my old answer which was clearly inadequate, and replace it with a new, clearer one).