The spell Tiny Servant (XGtE, p. 168) lets the caster create a Tiny creature out of an object:
You touch one Tiny, nonmagical object that isn’t attached to another object or a surface and isn’t being carried by another creature. The target animates and sprouts little arms and legs, becoming a creature under your control until the spell ends or the creature drops to 0 hit points. See the stat block for its statistics.
If a wizard casts Tiny Servant on a crossbow, would the crossbow be able to fire itself at an enemy?
In the statblock, the only action the servant can take is "Slam":
Slam. Melee Weapon Attack: +5 to hit, reach 5 ft., one target. Hit: 5 (1d4 + 3) bludgeoning damage.
A crossbow does 1d6 (hand crossbow), 1d8 (light crossbow), or 1d10 (heavy crossbow) damage, which is noticeably more than the Slam action. Moreover, the extra range of the crossbow would let the Tiny Servant stay out of harm's way during combat.
Would it be appropriate to let the Tiny Servant crossbow fire itself during combat?
Presumably, the Tiny Servant wouldn't be proficient in firing its own ammo, but the extra range and damage would be invaluable.
Best Answer
A crossbow probably wouldn't count as a Tiny object, so this wouldn't work either way.
According to the table on page 247 of the DMG for HP for objects of various sizes, a Tiny object would be something like a lock or a bottle, while a Small object would be a chest or lute.
A crossbow is far more likely to be comparable in size to a lute (and thus considered Small) than a bottle (or stein/mug as depicted in the book), so it would be an invalid target for the Tiny Servant spell. Even the Flying Sword in the Monster Manual is considered a Small creature.
If it did work, however, then you'd likely be in house-rule/homebrew territory.
The Tiny Servant spell gives you the stat block for the animated object. That's it. Anything beyond what is in that stat block is up to the DM to adjudicate.
Side note:
Take a look at the Animate Objects spell on page 213 of the PHB. The Tiny Servant is close-ish to the stats given to Tiny objects animated by Animate Objects. A crossbow, in terms of damage output, would fall more into the Small size category.