I quote an example I learned long ago.
the Izuna
The Izuna are miko in the southern parts of Japan, but instead of performing local duties of cleaning and maintaining a shrine, the Izuna-mikoto normally has to do the work of mediums.
Take note however, they don't believe in Gods, or God period, and that in classic mythology, it was rumored they all follow a common lifestyle of abstinence, and patience through meditation.
It is, if at all, they never do in fact worship anything.
I come to think of Reimu as a follower of such a path, who uses natural spirits (Aya, can be considered a youkai of the mountain, a natural occuring youkai guarding the mountain capable of manipulating wind), or even holy youkai (namely Yukari) to aid her conquest. Just as the old Izuna Mikoto did, they employed incantations that ask for the assistance of gods or god to come to their being, normally through possessing.
Many accounts recall they can summon spirits to inhabit them and talk in tongues, others say it gives them powers to do things normal humans cannot, such as lifting impossible weights. Izuna Mikoto can also be found in certain media concerning the making of holistic drugs, not unlike the Ayurveda treatment.
Izuna Mikoto have many accounts where they employed the strength of great deities, such as the Myo-oh (Bu Dong Ming Huang Da Pu Sha) the deity Bodhisattva that controls the element of fire, the fierce fourth heavenly guardian (as opposed to the Vaisravana); this being is said to make the Mikoto able to walk through fire, and rescue trapped individuals in burning houses and so on.
There are many accounts, talking in tongues is one such thing. In India, certain priests are known to have the ability to reach onto the human body and extract tumorous lumps from them and cure a human of an incurable disease, I reckon if Reimu is following the path of the Izuna, it would not be surprising that Reimu has the ability to be possessed by youkai and use their strengths, or employ certain means to utilize the youkai that surrounds her to do miraculous cures.