The scenario:

Fighter McFighterson has found himself between two opponents. He decides that he is best-off by grappling both of them so that his team can fire away. Maybe he'll pin them later with his Grappler feat, who knows?

McFighterson Succeeds and now has an enemy in each hand.

Can Fighter McFighterson make attacks against either (or both) of his two grappled opponents?

Either Unarmed Attacks, or using each of them as an improvised weapon against the other.

Related: What is an unarmed strike

Related: Enemies as Weapons

Yes, you can make as many unarmed attacks as you can utilizing your Attack action, any Extra Attacks, or any Bonus Action¹ against either or both of those grappled target.

If you want to attack them with unarmed strikes in the same turn that you have grappled them then you:

1. will need to have spent from your attack action using to successfully grapple each opponent .


2. Have them within reach.


3. Have a free hand per opponent.


4. 
   

   Be no smaller than one size smaller than your target (PHB page 195):

   
   
   

   
   

   When you want to grab a creature or wrestle with it, you can use the Attack action to make a special melee attack, a grapple. If you're able to make multiple attacks with the Attack action, this attack replaces one of them. The target of your grapple must be no more than one size larger than you, and it must be within your reach. Using at least one free hand...

   
   

   
   



5. Use any remaining extra attacks or bonus action attacks to make your unarmed strikes.

Note the rules governing unarmed strikes that needn't require a free hand as they aren't a weapon, or that the strike be a punch. You may strike either target with your available knee, foot, head, mutant 3rd arm, etc.

PHB Errata V1 DTD 2015:

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.”

If you want to attack them on the round after you have grappled them, then all attacks you can use during your Attack action and any bonus action attacks can be used to make unarmed strikes against those grappled opponents.

^{1 - For clarity, I understand that, and intend to communicate that: a PC may only use a bonus action to make this unarmed attack if they have some sort of ability/feature/item that would grant them the ability to make it. One cannot simply decide to use a bonus action and punch/kick/head-butt something. Thank you various reputable users from the comment section for your helpful advice and discussion.}

Yes, McFighterson the fighter can.

From the PHB errata:

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.”

Cut and dry. He can even multi-attack with unarmed attacks such as: headbutts, knee blows or kicks.

Weapon attacks for the fighter would require at least one hand free, but unarmed attacks now clearly have no such restriction.