Lots of GMs in Adventure League
Means lots of opinions ...
... and a range of rulings.
However, on running a public game, the wizards.com / AL website states:
Run a fun game! As the Dungeon Master, you help guide the players through the adventure, and facilitate the story coming alive. Make sure you follow the golden rule: Make decisions and adjudications during the game that enhance the fun when possible.
That's pretty straightforward and sets the stage for plenty of options for Druid Wild Shaping.
As long as the beast has a complete stat block in a resource from the AL content catalogue, the GM is free to rule as they see fit. These resources include the Player's Handbook, Monster Manual, Dungeon Master's Guide, Volo's Guide to Monsters, Xanathar's Guide to Everything, Mordenkainen's Tome of Foes, Elemental Evil Player's Companion, and Sword Coast Adventurer's Guide. Other books may be added in the future.
From AL FAQ v8.2 (2018):
Can I Use the Monster Manual?
Creatures with complete stat blocks found in the Monster Manual and other resources listed in the ALCC are available for use with class features such as Wild Shape, Beast Companion, and various conjure spells. As always, your DM is the final arbiter for the rules, such as determining whether or not your character has satisfied requirements such as those imposed by your class (such as the druid’s requirement to have seen the beast in question).
Keeping a record of beasts
From my experience, GMs generally allow you to provide a list of beasts you have seen when starting at level 1. Moving forward you will want to keep a list of creatures you've encountered, but most likely only more exotic beasts like dinosaurs will be questioned. If you are out in the wilderness make sure to ask your GM if you've run across any beasts. Or if there's one you'd particularly like to see ask them if you've seen this one. They may not list every beast you run across unless you ask. You will, of course, not be able to wild shape until level 2.
What if you don't have the stats for a beast?
Ask your GM. They will be glad to help you with that. You can take a picture of the beast's stats with your phone or make a photocopy for private use if the GM will let you.