Can I cast spells through my Eldritch Cannon?

Title says it all. Can I channel my touch spells though my eldritch cannon, just like you can with Find Familiar?

No. The Eldritch Cannon feature does not describe any means of doing that, so that's not a thing you can do.

You can use a Homunculus Servant for that, though.

Cool, thanks!

The other question is whether it's documented anywhere whether a Cannon can definitely be worn by its user or even someone else in the party. There's a bit of debate in my group whether it can. Some are of the opinion that it can indeed be shoulder mounted or used in one hand as a firearm.  Some say it can only be "carried" as you would a suitcase, and it would need to be set down before the cannon could fire or the person carrying it can take an action.

Perhaps the limitation is that it can either be carried, or it can be self-mobile, but not both?

Quote from MongooseGef >>

The other question is whether it's documented anywhere whether a Cannon can definitely be worn by its user or even someone else in the party.

The rules don't say it must take up a hand when carried. In the older Tasha's version, it just said the tiny version would be small enough to be held.

There is no official documentation of this. The closest is a quote from Crawford, concerning the Tasha's version, from https://www.inverse.com/gaming/dungeons-dragons-tashas-cauldron-of-everything-review:

I asked Crawford if the Artillerist Artificer — a kind of magical inventor who can build arcane cannons — could build their small Eldritch Cannon as a shoulder-mounted projectile weapon.
“That’s not the intended use, but there’s no rule against it," he said with a chuckle.

Personally, I'd rule it could be shoulder-mounted, or hat-mounted, or whatever.

Also, I think it's eminently clear that someone other than the artificer could carry an eldritch cannon.

This is the ruling that our group will likely use:

Since the description states that you decide whether it is carried or not, that would determine whether it is tiny (and carried) or small (and self-mobile). It could therefore not be both simultaneously, and would need to be re-built after a long rest or using a spell slot to change its form.

In tiny "carried" form it could be shoulder mounted and could pivot to target enemies. But if it's removed and placed on the ground, it would just sit there like any other dropped weapon.

In small "mobile" form the cannon moves around by itself (15ft) and can target enemies. If it's picked up and moved by a PC then the cannon can only fire in the direction the PC is facing (the PC might need to do a check if the cannon is using flamethrower or ballista mode, since it could jump out of their hands when it fires unexpectedly), and the PC cannot take an action that requires their hand while carrying the cannon. Encumbrance rules would apply, so a small creature may not be able to carry it very far.

Ah, alas - I am literally chatting with our DM as I type this. It sounds like he doesn't want me getting too OP right out the gate so he's gonna have me do a bunch of skill checks to determine whether my Artificer is able to invent a magical quick release shoulder turret mount for my cannon. Wish me luck!

The text is this:

Using Smith’s Tools or Woodcarver’s Tools, you can take a Magic action to create a Small or Tiny Eldritch Cannon in an unoccupied space on a horizontal surface within 5 feet of yourself. The cannon’s game statistics appear below. You determine its appearance, including whether you carry it or not (and your choice of legs or wheels, for the latter). It disappears if it is reduced to 0 Hit Points or after 1 hour. You can dismiss it early as a Magic action.

 

As I read it, there's nothing here that says that a Tiny cannon cannot move on its own or that you cannot carry a self-mobile cannon. There's also no rule that says it has to be carried in a hand, instead of say, on a platform or strap attached to your backpack, shoulder, or hat. As a practical matter, since the cannon only has 15' of movement, I always elect to carry it some of the time. It has no RAW weight so it can't really encumber you, though it would be fair to wonder if a small creature can carry a small cannon. I think it's perfectly reasonable to treat it as a tiny object you carry that can also move 15' from you.

Honestly I don't see this as OP in any way. Note that since the cannon is an object, not a creature, it cannot flank or take attacks of opportunity, and it cannot do anything unless it is commanded by your bonus action. If you elect it in small size, it can be used for cover.