All damage inflicted has a damage type.
Vulnerability is simply taking double damage from a certain damage type.
For the most part players don't have vulnerabilities to certain types of damage, there are a handful of ways they can temporarily suffer it.
D&D 5E doesn't distinguish between players suffering vulnerability because they're wearing different types of armour or anything similar.
Contagion and Hallow are spells which have the option to inflict vulnerability unto those affected by the spell.
Similarly Armor of Vulnerability is a cursed magic item that inflicts vulnerability on those who wear it.
It's more commonly found with monsters being vulnerable to certain types of damage.
But even then it's fairly rare.
A quick search of the System Reference Document PDF shows...
Earth Elemental, Mephits, Mummies, Rakasha, Shadows, Salamander, Skeleton, Treants and uh... Awakened Shrubs
All have at least one vulnerability.