In 5th edition combat, does a character using a ranged weapon like a bow or crossbow get to add their dexterity to their damage? Such as a rogue with a 15 Dexterity using a light crossbow, would they deal 1d8+2 damage?
Yes, a ranged attack with a ranged weapon would include your Dex bonus to both attack and damage rolls.
The Player's Handbook mentions this twice.
First in chapter 7 ("Using Ability Scores") regarding Dexterity:
Dexterity Attack Rolls and Damage
You add your Dexterity modifier to your attack roll and your damage roll when attacking with a ranged weapon, such as a sling or a longbow.
This is stated again in chapter 9 ("Combat") under the section "Making an Attack":
To make an attack roll, roll a d20 and add the appropriate modifiers. ...
The ability modifier used for a melee weapon attack is Strength, and the ability modifier used for a ranged weapon attack is Dexterity. Weapons that have the finesse or thrown property break this rule.
So yes, for a character with 15 Dexterity (i.e., +2 modifier), they would normally calculate the crossbow's damage roll as 1d8+2.
Note that this only applies to ranged weapon attacks. Ranged spell attacks usually don't use Dexterity.
Some spells also require an attack roll. The ability modifier used for a spell attack depends on the spellcasting ability of the spellcaster.