Whether it's a religious organization, a company, a neighborhood, a store, a family, or any other group that could have internal politics, I think the answer would be the same: if there's an explicit connection to a government process or a public policy then it's probably on-topic. Otherwise, the question is either off-topic or there's a more suitable SE site to try migrating it to.
Just to reiterate the very first line of the 'What topics can I ask about here?' page of the help center:
Politics Stack Exchange is for objective questions about governments, policies and political processes.
The 'governments' part is pretty clear, but 'policies' and 'political processes' and their own are unfortunately fuzzy. Would a company's hiring policies fit? Would the internal politics of your local HOA fit?
Though that one line isn't completely clear, the rest of that page seems to suggest that the focus of the site is on the politics of governments and the politics that influence governments:
Central to the idea of this site are the nuts and bolts of policies introduced by governments...
Processes are central to how legislation is made. Questions seeking to understand the rules and processes by which policy is made in various legislatures or ruling bodies ... are wholly on topic
Just to clarify, Macroeconomics is specifically on-topic. Most public policy questions involve economic matters
In just about any policy of substance, there are particular personalities that are central to its understanding, as well as demographic data about supporters and opponents of legislation...
...answerable using polls, punditry, and other verifiable and reproducible sources.
So, to use the Can the US Roman Catholic Bishops establish rules against the will of the Pope? question as an example, there are a number of different ways to ask about that situation that I think could be on-topic here:
- What authority does Vatican City have over bishops in the US?
- This is similar to the original question, however it's framed to focus on Vatican City's authority as a sovereign government over citizens of another sovereign nation, rather than the ecclesiological aspects of the Pope's authority as a religious figure
- What ways do bishops have to lobby the the US government?
- Can focus on how the internal politics will translate to attempts to convince the government to enact similar policies
- Do bishops have any official positions in the national/state governments in the US?
- Asks if the people doing the internal politics have any direct influence within the government already
- Are there polls showing how much influence bishops in the US have over voters' or politicians' behavior?
- Asks how the internal politics will indirectly affect the government by connecting it to how people who control the government behave
Each of those versions takes the original topic and explicitly tries to connect it to government processes or public policy. Framing a question in terms of things like the extent of government authority, how much influence a group has in a government, how a group affects voters' behavior, etc., will help push a borderline question over to being on-topic.
In short, a question about the internal politics of a religious organization could be on-topic here if it was framed in the right way. If the religious organization and the local/national government are tightly linked in some way (Vatican City, Iran, the Church of England, etc.), then questions about their internal politics often are questions about government processes. In places that are more secular and religious organizations have no official place in (most of) the government, then questions would only be on-topic if they focus on how the internal politics are affecting the larger political systems those organizations are in.