You can make unarmed strikes while Wild Shaped, but they do normal unarmed strike damage.
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 is now defined extremely clearly. On the plus side, 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 your unarmed strikes can definitely replace your natural attacks.
The downside to this is that your natural attacks definitely don't count as unarmed strikes, so you can only do your normal unarmed strike damage. This was also confirmed by Jeremy Crawford.
The weapon attacks of a creature such as a Beast are 'Natural Weapons' and each entry for those attacks found under the Attack heading for each beast stat block tells you what kind of attack it is.
For instance, a Claw and Bite attack count as a 'Melee Weapon Attack'. Nowhere in the description for these natural weapon attacks do they count as either Finesse OR Simple natural weapons. Those are weapon types attached only to weapons found on the weapons table of the PHB.
Some attacks, such as a spike or spit attack count as a Ranged Weapon Attack for purposes of abilities or skills.
In conjuction with Monk multiclassing? No weapon attack (the natural attack options each beast shape allows you) you could ever make in a Wild Shape form would count as Finesse or Light or Monk Weapons, but they do count as a weapon attack.
Alternatively a 'Unarmed Strike' can be a headbutt, punch, kick, or other forceful blow. I think a bear is capable of a forceful blow... don't you? If only from a headbutt or body check. Which would then use your Monk's Martial Arts damage die and any attached abilities if the form is physically capable of the feature.
Best Answer
Yes
When you are transformed you are using the stat block of the beast you are transformed into. The attacks a beast makes are melee weapon attacks. Using an example from the humble badger:
Generally any attack that is not a spell attack is a weapon attack. Do note that weapon attacks are not necessarily attacks with a weapon, which is relevant for certain spells and effects.