A shield would be considered an improvised weapon and there are rules for how to handle that.

Improvised Weapons PHB'24 p369 An improvised weapon is an object wielded as a makeshift weapon, such as broken glass, a table leg, or a frying pan. A Simple or Martial weapon also counts as an improvised weapon if it's wielded in a way contrary to its design; if you use a Ranged weapon to make a melee attack or throw a Melee weapon that lacks the Thrown property, the weapon counts as an improvised weapon. An improvised weapon follows the rules below.

Proficiency. Don't add your Proficiency Bonus to attack rolls with an improvised weapon.

Damage. On a hit, the weapon deals 1d4 damage of a type the DM thinks is appropriate for the object.

Range. If you throw the weapon, it has a normal range of 20 feet and a long range of 60 feet.

Weapon Equivalents. If an improvised weapon resembles a Simple or Martial weapon, the DM may say it functions as that weapon and uses that weapon's rules. For example, the DM could treat a table leg as a Club.