That's up to your GM, I'd say. The narrative of the AP seems to be a bit more rushed, though.
The short answer is no, but there's nothing stopping them from having been a student at the Kith before the AP begins.
That said there is a major caveat...
The important question is when they want to do this, and what their purpose for doing so is (i.e. do they want to become a Lion Blade).
If they'd like to have been a student there prior to the AP's first book, that's completely viable. Martella might have gotten a recommendation to scout a potential spy in the making from the school (before they were recruited by the Lion Blades).
If they want to pursue it during the AP, it's pretty much a no go with one huge exception. Books 1 to 3 all assume the PCs are being kept pretty busy in various parts of Taldor. However, Book 4 quite literally involves you going to the Academy and getting a crash course on being a Lion Blade. So if that's their true objective, allow them to plot their character in a way that will allow them to become a Lion Blade at level 10 (if you're willing to delay their level up to after the training) or level 11. Of course, actually taking the Lion Blade prestige is entirely optional; the PCs basically become honorary Lion Blades at the end of their training, no matter their class.
If they opted to study before the AP, then this can function as a neat homecoming moment, since their studies were interrupted. If they didn't, then it could be a dream come through via a very unconventional workaround courtesy of their royal connections.