From Carson Sinclair’s card, we know he can give another investigator within range 1 extra action. Say a player (for example, Rita) is in range of him and is stunned— can that player use their one movement action then Carson’s extra action to move as well?
I don’t suppose so because there aren’t any exceptions on Stun, right?
That's a great question. The solution is all in the wording of Stunned and Carson Sinclair's Investigator Ability:
QuoteStunned: You cannot perform more than a single action during your turn . At the end of your turn discard this card.
QuoteAction: Another investigator within range may perform 1 action. Activate this ability only once per round.
When Carson Sinclair uses his ability, it is 'Carson's Turn.' Being Stunned restricts you only on your Turn.
In your example, when Rita used her one action during 'Rita's turn' she would be free of being Stunned at the end of her turn, and then Carson uses his ability during his turn to give Rita the extra action. There is no issue here.
It's the other way around that would make people think twice. Can Carson use his ability on an Investigator that is Stunned (that has the Stunned Condition)? Yes. Stunned only affects an Investigator during their own turn. Carson uses his ability during his turn, so grants an extra action to the other Investigator outside of the Stunned Investigator's own Turn.
(Also, be careful not to confuse an Investigator's 'Turn' with a 'Round.' An Investigator (normally) performs two actions during their Turn, while a Round consists of the whole Investigator Phase and the Mythos Phase.)