Canonically, a player can probably demand a Fate Point for it not being a full moon, darn the luck.
When faced with a lycanthrope PC, a GM will have to consider how often she's willing to stage stories near the time of the full moon - and, if she is willing, how much of a restriction the Human Form (Involuntary Change) really represents. (However, deciding that the full moon is not forthcoming may be worth a compel every time it's relevant.)
-- Dresden Files: Your Story p. 79, "Types and Templates"
When you make a character you assume you're playing them at full power - you're not expecting the GM to just suddenly declare Karrin Murphy catches the flu, can barely see, and is housebound, at least not without some kind of chance to avoid it or compensation for accepting it.
Lycanthropes have a codified version of this in their Human Form (Involuntary Change) restriction, which reduces but does not eliminate the cost of their enhanced strength and toughness by restricting it to only operating during the handful of days when the moon is fullest. As the section says, when you play a lycanthrope it's with the expectation you'll be operating at least some of the time in that enhanced form, but the restriction is there to make it possible that you're not.
Exactly what that means for the game will need to be discussed with the GM during game creation, but a good starting point for that discussion is that when you're doing investigations on your own initiative it's when the moon is sufficiently full to grant you your powers, but enemy action or unfortunate timing can result in you being in an important scene without a full moon. And usually when enemy action or unfortunate timing puts you at a disadvantage, that's worth a Fate Point. Though it can also be an avenue for the GM to introduce time-based complications into an investigation - there's only so much time for you to track something to its lair before the full moon passes and you're not operating at full power.
It's probably not the case that this discussion is going to come out to you having to pay Fate Points during the game to be at full werecreature strength. After all, you're already effectively paying Fate Points every game (with the Refresh cost) in order to have that strength in the first place.