It's not even tell they actually joined the League in their lore. I guess you can say Shaco did because he is crazy and wants to kill people, but Udyr?

I am not sure why Shaco joined the League, but i know that Udyr joined the League to control his powers which sometimes took control over him. Hope I helped!

Shaco's reason is purposely kept mysterious, as questioned by his lore.
Udyr.... I dunno. Maybe to channel his primal fury into something productive?

remember he was a monk at one point, justice prehaps?