One possible reason could be feeling ostracized. In the Naruto world, being different is often not easily accepted. As a female Naruto, she might face even more scrutiny and prejudice. Maybe she's tired of the constant expectations and the way she's treated in Konoha, so she decides to run away to find a place where she can be herself.
She could encounter other outcasts. There are likely other ninjas or people in the world who have left their villages for various reasons. By meeting them, she could form a new group, a sort of 'family' of misfits. They could travel together, sharing their skills and stories, and maybe even start a small community away from the traditional ninja villages.
Maybe in the fanfic, Naruto has faced too many political intrigues within Konoha. He has always been loyal to the village, but if he constantly encounters backstabbing or unfair treatment from some of the village's leadership or clans, it could lead to his weariness.
Maybe he's running away from the pressure in the village. There are always high expectations on him, and he might feel suffocated and need some time alone to figure things out.
One consequence could be that his friends worry about him a great deal. They might go on a search for him, which could put them in danger as well. After all, they care about Naruto deeply and won't just let him disappear without trying to find him.
Shikamaru might be sent to find Naruto. He's known for his intelligence and strategic thinking. The village elders might trust him to be able to track Naruto down and either convince him to come back or at least find out his intentions. Shikamaru also has a certain respect for Naruto and would understand his motives to some extent.
In this fanfiction, Naruto might leave because of the political situation in Konoha. There could be factions that he doesn't agree with or that are causing unrest. Naruto, being someone who values peace and unity, might find it difficult to deal with the internal politics and decides to leave in order to find a more harmonious place or to figure out a solution away from the chaos.
He might meet some of his distant Uzumaki relatives. There could be elders or other Uzumaki clan members who have valuable knowledge about his family and their special abilities.
Perhaps he is running away from danger in Konoha. There could be a threat that he believes he can't face directly in the village, and he thinks Whirlpool might offer some sort of sanctuary or a way to become stronger in relative safety.
Maybe he's running away from some kind of danger or threat in his home village. There could be a group or an individual after him, and he believes that Kumo would be a safer place. It could also be related to some political unrest in Konoha that makes him feel like he has to leave, and Kumo seems like a viable option.