There are probably a couple issues with reincarnating someone from Earth.

1) Knowing that they exist, who to reincarnate
2) Knowing their true name or having part of their body
3) What system were they using for their afterlife? If they dissipated, then there isn't much you can do. And if they went somewhere guarded by some deity who allows no other deities before themselves...it might be hard prying them out of that deities grasp.
4) On the other hand, grabbing a Buddhist or similar might be possible...IF you can catch them between reincarnations...and there's the rub, if they are currently incarnated you have to get them out of their body and onto the Astral plane....

As a rule, most clerics don't have direct access, they go through an avatar to whom they are dedicated, and who has access. And in fact that avatar may go through another avatar. It sort of depends on how important you are, how important the avatar you are talking to is, and what you are doing.

Think of it as a "resources" approval process. Different levels of clerics are authorized to use so much god mana (beyond their own local supply) which in all likelihood is not from the GP but from their avatar. That avatar has so much mana he/she can give out, personally, and they can pull up to a certain amount out of the GP (think of it like a mana budget) Higher up avatars have bigger budgets/can approve more mana usage.

This actually is discussed in book II, particularly in regard to Saint Hilda of Rivenrock.

As an FYI the priests that feed her mana, and which she returns mana to are called her Illuminaries (because they light her up) in turn she is an Illuminary to an even higher up Archon.

And I would point out that most priests can swap between multiple Illuminaries, it depends on who they prey to, who they have established relationships to. For example, they can call on different saints for different situations, or different angels for different situations. They have their default "Patron" saint (some of whom are actually angels--think how the Seraphim have "Saint" analogues in Christianity)

Typically, most priests would use a Saint or other avatar's assistance to do a reincarnation. In theory, depending on the religion, a high priest or "future" saint, might be able to do it on their own. Remember it takes verified "miracles" basically done on their own to get a person canonized.

Think of Canonization as being something like a religious version of Demon Binding, only it's creating an Avatar (or Saint) out of a human spirit that has exited the body.

I doubt it would work, as it requires the consent of the deity (or avatar)...as do most of the more powerful clerical rituals.

And the person has to be dead and not have a body, demons have bodies (sort of)

I suppose it could be possible for an evil deity to want to get demon avatars...but I'm not sure that they'd need to do canonization, they could presumably do something else to link them up to the GP and the "system" since Demons and Angels are very similar constructs.

In fact, in many pantheons, demons are just rogue angels that have broken away from the deity....

So, not being a priest, and having never been attached to a deity, I can't say, but presumably some of the canonization spells involve forming the link to the godpool and the deity network. You'd do those rituals, but wouldn't do the incarnation rituals that create the "body" as the "body" already exists.