It might make some sense to use the word principal (as has been independently suggested elsewhere on this page).
This has many related meanings; the one I'm thinking of is the legal/diplomatic usage in which someone sends an agent or representative to act on their behalf. The person for whom the agent acts is their principal. Wiktionary defines this meaning as a:
person that authorizes another (the agent) to act on their behalf
And Apple Dictionary as a:
person for whom another acts as an agent or representative
That's not exactly the meaning you want, of course, but I suspect it's close enough that most people would understand it fairly easily.
(What may make it more appropriate is that some of its other noun meanings include the most important, most senior, or most responsible person: the chief executive of a university or college, the owner of a business, the highest-ranked dancer in a ballet company, the leading player in a section of an orchestra, and so on. And of course its adjective meaning is ‘primary; most important’, which also seems relevant.)
I would suggest source. I don't think avatars are common enough that we have a specific word for their originators/controllers. I certainly can't think of a word which is so specific that it wouldn't be used of non-avatars. I think the most important things that Vishnu does for Rama are that Vishnu created Rama, and that Vishnu controls all of Rama's actions, and that Vishnu "is" Rama's mind/Rama's soul is Vishnu's soul, etc.. "Source" captures the first meaning and sort-of captures the others.