Our cleric (of Apollo) wants to cast spells through his weapon-
Using it as a weapon focus is fine, particularly if the cleric was of the War domain. Personally I wouldn't put it in Apollo's wheelhouse unless it were a bow, but you can do what you'd like. As others have said, it's not going to be broken because the books give an example of a shield emblem in the description of a holy symbol on PHB 151.
specifically, he wants to temporarily charge the weapon with a spell, which will be cast on the enemy when he attacks with it. Are there existing rules that could support this? If not, what could I put together that would still be balanced? The party is level 11 at the moment.
That's a horse of a different color.
The "rules as written" version of this would be to say, "Take two levels of Paladin and gain the Divine Smite ability," although the character would probably want to take five levels of Paladin to get Extra Attack if he's so focused on combat. If you want to do it as a reward, you could just grant the ability outright or let the player spend their ASI at level 12 for it, but it's not a minor ability.
Another option would be to create a weapon that lets you cast a specific spell X times per day, or perhaps a weapon that would allow you to use Divine Smite 3-5 times per day.
If we're looking for a more direct equivalent, I don't believe it's in 5e, but
3.5e had a Spell Storing weapon modifier:
A spell storing weapon allows a spellcaster to store a single targeted spell of up to 3rd level in the weapon. (The spell must have a casting time of 1 standard action.) Any time the weapon strikes a creature and the creature takes damage from it, the weapon can immediately cast the spell on that creature as a free action if the wielder desires. (This special ability is an exception to the general rule that casting a spell from an item takes at least as long as casting that spell normally.) Once the spell has been cast from the weapon, a spellcaster can cast any other targeted spell of up to 3rd level into it. The weapon magically imparts to the wielder the name of the spell currently stored within it. A randomly rolled spell storing weapon has a 50% chance to have a spell stored in it already.
Note that the ability was restricted to melee weapons, so spell range wasn't a consideration of the design. 3.5e classifies it as a +1 equivalent, which is the weakest category of weapon enhancement. As such, a magic weapon with an equivalent ability in 5e would not be too far out of line, especially at level 11. I would probably make it a +0 or +1 magic weapon
at most. A 5e version would probably also require the spellcasting to cost a reaction because casting spells is harder in 5e, but that's somewhat fiddley for a homebrew item.
From the default Cleric spell list, I would include:
Cantrips: Sacred Flame,
Spare the Dying, Toll the Dead
1st: Command,
Cure Wounds, Guided Bolt,
Healing Word*, Inflict Wounds,
Sanctuary*
2nd:
Aid, Blindness/Deafness,
Gentle Repose,
Lesser Restoration,
Protection from Poison,
Warding Bond
3rd: Dispel Magic (creature target only),
Feign Death,
Life Transference,
Remove Curse,
Revivify,
Sending,
Tongues,
Water Walk
Anything in italics would be technically legal, but would be really narrow in use. Many of the cleric spells are willing creatures only or are beneficial to the target. It's a pretty limited list for Clerics. Note, however, that
other characters in the party could charge the weapon.
Vampiric touch was the 3.5e spell that was classically paired with the weapon.