Familiars have the statistics of the form they take
The text of the find familiar spell says, in part:
the familiar has the statistics of the chosen form
Inasmuch as equipment is part of the statistics for some forms (e.g. the sprite) those forms are unambiguously guaranteed to start with that equipment which is part of their statistics.
The monster manual provides the basis for equipment being part of a creature's statistics:
A monster's statistics, sometimes referred to as its stat block, provide the essential information that you need to run the monster.
Note that e.g. a long sword attack in a monster's stat block doesn't just mean it can use a long sword; it means it has a longsword which it uses in combat with the following rules. You can disarm such a creature, but e.g. when a module says the PCs are attacked by cultists, those cultists are assumed to have scimitars and make scimitar attacks until they don't.
Textual support for a monster having the equipment listed in their stat block:
A monster that wears armor or carries a shield has an Armor Class (AC) that takes its armor, shield, and Dexterity into account. Otherwise, a monster’s AC is based on its Dexterity modifier and natural armor, if any. If a monster has natural armor, wears armor, or carries a shield, this is noted in parentheses after its AC value. (MM, Introduction)
This is clearly the case for the Sprite, which has an AC of 15 (11 from its leather armor, plus its +4 DEX bonus). That its AC is exactly what its armor and relevant Ability would imply suggests that it actually has and wears the listed armor. It's not explicit rule text, but rather a direct application of the relevant rules if and only if the monster actually has and is wearing the armor.
A stat block rarely refers to equipment, other than armor or weapons used by a monster. A creature that customarily wears clothes, such as a humanoid, is assumed to be dressed appropriately.
You can equip monsters with additional gear and trinkets however you like, using the equipment chapter of the Player’s Handbook for inspiration, and you decide how much of a monster’s equipment is recoverable after the creature is slain and whether any of that equipment is still usable. (MM, Introduction)
Also probably not the specific rule text the OP is seeking, but explicit mention that you can equip monsters with additional gear and trinkets (additional to the standard equipment in the stat block, presumably) suggests that the monster is equipped with the things listed in the stat block.
Obviously a DM can override this at will, but there isn't any converse rule verbiage-- there aren't statements, or even implications, that a monster does not have the equipment listed in their stat block. The only RAW evidence which exists supports the idea that monsters do, by default, have the equipment their stat blocks describe.