A grapple attempt is an attack - a special melee attack:

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.

Because it's an attack, it can target an illusory duplicate from the mirror image spell:

Each time a creature targets you with an attack during the spell's duration, roll a d20 to determine whether the attack instead targets one of your duplicates.

What will happen if the grapple attempt targets a duplicate? Will it automatically be destroyed if I win the contest? What will happen if I lose the contest?

Nothing happens to the image. The grapple attempt just fails.

The pertinent mirror image rules:

A duplicate's AC equals 10 + your Dexterity modifier. If an attack
hits a duplicate, the duplicate is destroyed. A duplicate can be
destroyed only by an attack that hits it. It ignores all other damage
and effects. The spell ends when all three duplicates are destroyed.

And the grappling rules:

The target of your grapple must be no more than one size larger than
you and must be within your reach. Using at least one free hand, you
try to seize the target by making a grapple check instead of an attack
roll: a Strength (Athletics) check contested by the target’s Strength
(Athletics) or Dexterity (Acrobatics) check (the target chooses the
ability to use). If you succeed, you subject the target to the
grappled condition.

The grapple is an attack (specifically a "special melee attack") but not one that can hit. As such, it would have no effect on the mirror image - subjecting it to the Grappled condition does nothing.

That's further supported by the fact that you cannot either win or lose this contest, as the mirror image does not have a Strength (Athletics) or Dexterity (Acrobatics) check to contest against. By design, the only way to hit (and therefore dispel) the mirror image is to target AC.

For clarity, the chain of events would be as follows: You attempt to grapple a wizard who is currently under the effects of Mirror Image. You roll your Mirror Image check. If the roll is in your favor, then you correctly target the wizard, and the grapple attempt proceeds as normal. If it is not in your favor, you wind up targeting an image instead. The mirror image is unaffected by grapple, and therefore the grapple attempt autofails.

Further support for this position available in the answers to this question here.

Even more support can be found in this Sage Advice Compendium (page 11, right-hand colum)

Does a grapple or a shove trigger the Tempest cleric’s Wrath of the
Storm or a Battle Master’s Riposte? The answer to both questions is
no. The grappling and shoving options (PH, 195) don’t result in a hit
or a miss.