Q: How can Lea wield the Keyblade?
A: Lea’s journey to become a Keyblade Wielder is one that developed over the course of KH3D, even though it wasn’t explicitly mentioned until close to the end. Lea knew that, in order to bring Roxas and Isa back, he would need a new power to do so. In that regard, he set out to find Yen Sid and ask to become a Keyblade Wielder.
Yen Sid was quick to tell him that it wasn’t an easy path but Lea was determined. Off-screen, Yen Sid or Mickey performed the Rite of Succession with Lea and then immediately set him out to train with a Keyblade. Merlin and the Three Fairies created a training area for Lea where time flowed differently in order for him to receive hastened training on how to properly wield one.
Even with this training though, a Keyblade still needed to choose Lea as its master, and, because one hadn’t yet, he couldn’t summon a Keyblade of his own. The most important requirement to wielding a Keyblade is that the owner must require a strong heart (regardless of whether this heart is “good” or “bad”, it must be strong). Lea’s desire to regain certain things was key in determining the strength of his heart, but it wasn’t until he rescued Sora did he truly prove the strength of his heart.
Once he had, a Keyblade chose him, and he was able to summon it. Thus, he became a full-fledged Keyblade Wielder.
Q2: Couldn’t seeing Isa have been the trigger for the Keyblade choosing Lea?
A2: It’s really up to personal interpretation what exactly caused the Keyblade to choose Lea. All we know for certain is that before rescuing Sora, Lea couldn’t wield, but he could after doing so. Clearly the answer is within that event. If someone wants to believe it was seeing Isa that did it (strengthening Lea’s resolve to save him) they could. Just like someone could believe it was the act of rescuing Sora (or even both). It’s up to interpretation and either interpretation is correct. All that matters is that because of those events, the Keyblade finally chose Lea.
Q3: In Birth By Sleep, Lea grabs Ventus’s wooden Keyblade. Couldn’t that mean Ventus unknowingly performed the Rite of Succession on Lea?
A3: In order to perform the Rite of Succession one needs to be a Master (or have the sufficient power of one). Ventus was never at that level in Birth By Sleep, and therefore couldn’t perform the Rite of Succesion, even unknowingly. Furthermore, the wooden Keyblade isn’t an actual Keyblade, and therefore, it too, can’t contribute to a successful Rite of Succession. Lea playing with the wooden Keyblade was likely just a foreshadow to the events of KH3D, where Lea wields one for real, not an actual answer as to how.