Appropriate Word or Phrase that Means "Determine Whether an Action is Permitted"

From the question:

At the moment, I'm clunkily describing this as "determine permission", but that feels inaccurate because one might determine that something is not permitted.

If you only want to ascertain the one state, then, changing the suggested phrase as little as possible, try the following:

verify permission:

[Merriam-Webster]
// 1 : to establish the truth, accuracy, or reality of
// verify the claim

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I will note, however, that the wording used in the actual example sentence is slightly awkward, because, in that context, the original determine does seem like the best fit—with a slight rephrasing:

Its members alone determine the application's approval or denial.

It's not the members who would verify a particular state. All they do is make a decision.

Additionally note that requests aren't said to have permission or not. People and actions are given permission, not requests. Requests are only approved or denied.

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A more natural sentence, based on the same scenario, would be something like this:

✔ Contact the committee secretary, who will verify if you have been given permission [to …]

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However, that aside, verify is the more accurate verb for checking or confirming the state of something specific.

Provide ratification should work.
You could also try 'allow'/'authorise for'/'provide authorisation for'.
Hope it helps