dnd 5e – How to tame a wild animal without being / knowing a beast master?

One of my recent DnD sessions, one of the group members (rouge) domesticated a wild turkey. Since then, the rest of us have also wanted a companion / pet (for example, Blood Hawk, Bat ect). Are there certain rules? Or is it completely from the DM? Because with the turkey they domesticated it in a roll.

Keep in mind that none of us are Rangers or beastmasters.

And we're playing for the adventure league (DM usually decides to use the rule or not)