It doesn't seems to be possible
A search in the Pathfinder: Open References application, gathering most of the sources available regarding spellbooks will just explain you that a spellbook is a... book... in which you write spells.
As it's not very indicating, I looked up how to add spell to spellbook. The rules regarding the addition of spell to a spellbook are more informative: they only exists for prepared arcane casters and are stated in their class abilities.
So RAW, nothing exists to let a prepared divine caster write down spell in a spellbook (since they don't possess it as a class ability).
From a flavor point of view
From a lore point of view, divine casters do not know any spell. They pray to an higher being to cast spell through them, their faith acting as a beacon and channel for the divine energy.
They can indeed write scroll, but it stands to reason if you consider a scroll to be a readied spell fixed on paper. Once it goes off, the spell disappears. Divine casters are able to capture the spell being casted through them in a scroll, but they do not "learn" that spell, merely fix its energy on paper.
From a design point of view
I think you need to factor that having a spellbook is a liability to arcane casters: it can be taken away, damaged or destroyed, which leave the caster without any spells. This drawback does not apply to divine casters, as your deity will probably always be here to support you (unless you're a cleric of Aroden). Thus, I think from a design point of view, giving the opportunity for divine casters to permanently write down what's essentially miracle in a spellbook would tip the balance.
The mnemonic vestment actually allows a spontaneous caster to expand their known spell list. Unless you need a wide array of spells you don't know (as in, very wide), you can probably rely on a few scroll tucked up your sleeves to access a few essentials you don't want to bother learning.