It does not require surprise in AL play.

The original version of the bugbear was published in Volo's Guide to Monsters and required surprise:

If you surprise a creature and hit it with an attack on your first turn in combat, the attack deals an extra 2d6 damage to it. You can use this trait only once per combat.

However, this version of the bugbear was deprecated with the release of Mordenkainen Presents: Monsters of the Multiverse¹, and does not require surprise to use the Surprise Attack trait:

If you hit a creature with an attack roll, the creature takes an extra 2d6 damage if it hasn't taken a turn yet in the current combat.

This most recent version is the version of the race that is eligible in AL play.

^{1 The table contents for Volo's Guide on DND Beyond bears the banner: "This book contains Legacy Content! This means that the contents within don’t reflect the latest lore, mechanics, or the current state of Dungeons & Dragons. For more information, see Legacy Content."}

It no longer requires surprise in AL play.

The original version of the bugbear was published in Volo's Guide to Monsters and required surprise:

If you surprise a creature and hit it with an attack on your first turn in combat, the attack deals an extra 2d6 damage to it. You can use this trait only once per combat.

However, this version of the bugbear was deprecated with the release of Mordenkainen Presents: Monsters of the Multiverse (MPMM)¹, and the most recent version of the bugbear from MPMM does not require surprise to use the Surprise Attack trait:

If you hit a creature with an attack roll, the creature takes an extra 2d6 damage if it hasn't taken a turn yet in the current combat.

This most recent version is the version of the race that is eligible in AL play. From the DDAL Player's Guide v. 13.0, sidebar on page 1:

WHAT RULEBOOKS SHOULD I USE?

You can use non-optional rules found in the Player’s Handbook, the Basic Rules, and all the books listed below. If a rule has been reprinted in a newer resource, you must always use the latest printing.

• Fizban’s Treasury of Dragons (FTD)
• Mordenkainen Presents: Monsters of the Multiverse (MPMM)

Naturally, Volo's Guide does not appear on the list.

^{1 The table contents for Volo's Guide on DND Beyond bears the banner: "This book contains Legacy Content! This means that the contents within don’t reflect the latest lore, mechanics, or the current state of Dungeons & Dragons. For more information, see Legacy Content."}