I actually don't think they do - drones don't have the full set of actions a regular character does, and only get granted specific actions from the Limited/Expert/True AI class features; none of these provide a reaction.
Reading those sections on pages 75 and 76, Limited AI, Master Control, Expert AI, True AI, seem to suggest that drones don't have a reaction.
But if you read Shock Wave on page 79, you will see that it does mention reactions for the only time.
Shock Wave (Ex)Whenever your drone is reduced to fewer than 10 HP, it
immediately shuts down and unleashes a shock wave
of electricity as a reaction (even if it is deactivated or
destroyed by the attack).
The drone has to wait until level 11 to get this. Does this mean they get reactions from level 1? It seems more like an exception.
Looking at page 248, Reactions, we are told
You gain access to most reactions through feats, items, andclass features, but an attack of opportunity is a reaction that is
universally available to all characters.
Is a drone considered a character? Does it have a character sheet :) It has a section in the Player Character Folio.
So, down to the GM if a drone can take a reaction. I might say that a Combat Drone could take AOO but otherwise no.
CreatureA creature is an active participant in the story or world. This
includes player characters (PCs), nonplayer characters (NPCs),
A monster is a nonplayer character. In general, monsters are too
strange or unintelligent to be player characters, or are prevented
from being them for other reasons. A monster might be a player
character’s opponent or ally, or serve any other role.
A drone is considered a construct with thetechnological subtype for the purposes of spells and effects
Like most constructs, a drone cannot recover from damage on
Drones are constructs with the technological subtype. That makes them creatures. It's either a PC (nah..), NPC (maybe..) or a monster (maybe..) or something else not worth mentioning (dubious). Since monsters are NPCs, a drone is probably technically an NPC, and therefore a character.