It seems the user control for register page is pages/register.ascx, but it has no code-behind, so I don't think I can add my logic using code-behind.

But, I could perhaps add it to the stored procedure that gets called when a user registers on YAF. Is this the best approach? If yes, then what is the name of the stored procedure? I see a stored procedure called dbo.yaf_user_save, but not sure if this is the right procedure.

If you like to modify the code Behind file you need the latest Source Package

https://github.com/YAFNET/YAFNET/releases/download/v2.2.4.3/YAF-v2.2.4.3-Source.zip 

There is no stored procedure that is only called during the registration. The only way is via the Code.