The company Spin Launch is doing someone not that dissimilar to this, except they're not using explosives to shoot it up, but are accelerating it in a circle until it gets the right speed

A more practical method would be some type of mass accelerator. A canon implies very fast acceleration. We can shoot garbage in to orbit like that. But if we are puting delicate stuff in orbit, we need to spread the acceleration over time.

https://en.m.wikipedia.org/wiki/Operation_Plumbbob

The only time we might have sent an object in orbit via an explosion

With a cannon you can’t go much faster than 4-5x the speed of sound, unless the space in between the payload and the explosive is filled with a lighter than air gas like helium. Like in a light gas gun. But even then the acceleration needs to be extremely high since the only thrust the payload gets is at the beginning. So it would destroy most things especially squishy humans. A rail gun would be a better solution and is in theory possible, since it can accelerate the entire length of the rail. If you just make it long enough you can accelerate to any speed in a vacuum at survivable G-forces. But the end of the barrel needs to be in space in a vacuum as well other wise the payload will explode the moment it exists the barrel and hits atmosphere if it’s going fast enough to reach the moon. Though the space craft still needs it's own thrust system to decelerate, otherwise it will just loop around the moon and fall back to Earth.

As long as the force of the launch is enough to reach escape velocity, I guess so. But since the acceleration would be immediate, I guess a human occupant wouldn't be able to survive the immense g-force generated.

It's actually a more interesting solution for doing things the other way around: have some kind of mining operation on the moon and use a "cannon" (most likely a rail gun) to accelerate payloads from the surface of the moon into Earth orbit or even down to the surface.

But yeah, even that one wouldn't be for squishy payloads such as people.

PS: That said somebody else is pointing out existing experiments were the object is accelerated around and around by a mass acellerator and then release at the appropriate speed, which might work for human payloads.