Can a Druid/Monk who is Wild Shaped perform Unarmed Strikes?
Yes, you can make unarmed strikes while Wild Shaped. From the errata for the PHB:
Melee Attacks (p. 195). The rule on unarmed strikes should read as follows: “Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike: a punch, kick, head-butt, or similar forceful blow (none of which count as weapons). On a hit, an unarmed strike deals bludgeoning damage equal to 1 + your Strength modifier. You are proficient with your unarmed strikes.”
An unarmed strike can replace any attack with a weapon. From the Monster Manual, page 10:
The most common actions that a monster will take in combat are melee and ranged attacks. These can be spell attacks or weapon attacks, where the "weapon" might be a manufactured item or a natural weapon, such as a claw or tail spike.
So can definitely replace your Wild Shape attacks with unarmed strikes.
Although it seems powerful, it would make sense to me that any creature would have the ability to perform an unarmed strike since an unarmed strike is not simply limited to a punch.
Going off of RS Conley's answer, an Ape has the ability to perform a "natural attack" with its fists, but the Ape would also have the ability to perform other unarmed strikes such as a kick, headbutt, body slam, different kinds of attacks with its fists such as a karate chop, etc that could be fueled by the Monks class features. Even as something like a Horse, you can still headbutt, stomp, charge, kick, etc.
This also makes sense to me objectively since a monk perfects melee by achieving mastery of the human body, and a druid has a lot of instinct, intuition, and/or experience driving their use of a creature's body that allows them to use it just as effectively as their human form (implied since druids do not have any penalties while in beast form).
If you are talking about the unarmed strike mentioned on page 195 of the PHB the answer is no unless the wild shaped form is a humanoid like a Ape (CR 1/2). Even then the Ape has a superior attack with his Fist.
The exact wording of unarmed strike is
When you are unarmed, you can fight in melee by making unarmed strike, as shown in the weapon table in chapter 5.
Looking at the weapon table we see
Unarmed Strike Cost:nil, Dmg: 1 bludgeoning, Weight:nil, Properties:None
It is clearly representing a humanoid ability to punch or kick. Not claws, hooves, or any of the other attacks that animals are capable of performing.
I sense there is something more to this question. But since I don't know what that is I will end my answer with the following.
From a in-game standpoint all natural animal strikes are unarmed in that they are attacks that are intrinsic to the creature and not as a result of a tool, like a sword, or a supernatural power. In this sense the Bite attack or the Claw attack of a panther is an unarmed strike using a different set of stats then the ones found on the weapons chart.
Looking at the Ape again we see that the Fist is clearly a unarmed strike of the this type. But the second type of attack is Rock which is not an unarmed strike of any kind as it involves the ape picking up a rock and throwing it. So a druid, wild shaped as an ape, is attacking unarmed when using the multi-attack with two fists.
The Druid/Monk Multi-Class character
In my opinion the literal reading of the rules makes my answer no. The unarmed strike that various PHB abilities refers to is specific weapon listed on the weapon table in Chapter 5 of the PHB. While various animals are certainly attacking unarmed they are not attacking with that weapon which is limited to the PHB races of humanoid form.
However if you want to go with the common sense idea that a animal's natural attack is an unarmed strike. Then I would consider the following:
The main issue is the increased damage for unarmed strike. A normal human does 1 point of damage. The monk starts off with 1d4 and eventually increases to 1d10. I would say this does not apply while in a wild shape. The idea of increased damage is the monk overcoming the limitation of human punches and kicks through skill and training. A limitation that animals with claws, fangs, and hooves do not possess.
If the referee feel strongly that a monk training would increase the damage of an animal's natural attack then he needs to figure out a way to stepping up the attack die in a progression similar to the monk table. I feel that the this was not the designers intent.
As for the other abilities of a Monk, I would look careful at the description of the ability and see how the use of humanoid hands are involved. For example I don't see any particular issues with Flurry of Blows, however Deflect Missiles a Druid in a wild shape without hands would not be able to make use of the ability to catch an arrow, deflection would be unaffected. If the feels that it important to retain this ability he could add a house rule that it transform into the ability to catch the arrow with form's mouth or other suitable appendage like a tentacle if applicable.
I would say yes, the attack action of beast is unarmed but I would further say the bonus attack is based off of the monks abilities and thus use the monks unarmed attack damage (d4 and up) as mister dood answered animals can also use other parts of their body to attack like a human would like a cat can swipe with its front paws or pummel you with its hind legs while on its back.
In the marital arts entry under monk class it says "when you make an attack action using unarmed strike or monk weapon you can use a bonus action to perform an additional unarmed strike" it also states "you can use a D4 instead of the attacks damage dice" and in wild shape for druid class it states you retain any features granted by class or race whilst in beast form. Further more consider the spell alter self and the natural weapons component, I would say a druid using wild shape is closer to that than 'equipping'claws or teeth etc
I would say yes in most cases. It might depend on the creature you wild shaped into (an ape or bear, for sure; a snake, probably not). The beast's natural weapons would not count as unarmed, nor as a monk weapon. You could however make the unarmed strike (kick, headbutt, punch, karate chop, etc) so long as the beast's anatomy allowed for it. Some beasts could also wield a monk weapon (such as an Ape, and arguably a bear - though your DM might disagree with the bear). Note - your hit and damage modifiers would be based on the beast's dex/str, not your natural form's. (unless you are also a warlock who can use Chr for attacks, which is maintained while shape-changed).
Example 1 of your turn (after being shape-changed): action - attack with staff (if your beast has the anatomy to wield it), or unarmed strike bonus action - unarmed strike (or flurry of blows).
Example 2: Action - multiattack: bite, claws. Bonus action - n/a (unless you have some other class feature or feat that fits here)
For Flurry of Blows it would seem that most beast attacks are listed as 'Melee Weapon Attacks' which of course, wouldn't count as either unarmed or 'monk weapons'. That would seem to indicate no.
I'd say a fair compromise would be to have some way of selecting a single animal form that can have access to monks abilities. This thematically would reflect the training and dedication required for a monk. You would re-class that form's attack(s) as a 'monk weapon' and give them an additional 'unarmed strike' equal to the monk damage dice.
This also seems fairly balanced. A druid's wild shape should have access to roughly as damaging of an attack as a monk of similar level wielding a monk weapon. And, the damage you are getting from the unarmed strike is just what a monk could get access to. The most OP thing I can see is a druid 2/monk 1 morphing into a blink dog (1/4 cr), getting the dex from a blink dog (17!) and having a high wisdom. You'd get 22hp, an AC of 13 + wis, a bite with +6 attack doing 1d6 + 4 dmg followed up by an unarmed +3 1d4, and be able to teleport (it isn't flying or swimming! ... that's gotta be a typo)