First, let us define the problem: Santa wants to deliver presents ballistically, launching them from a cannon to descend down the recipient's chimney, yet not leave a crater at each present's intended destination. Santa's presents - at least those that are traditionally deliverable - share one characteristic: they are human portable, meaning that they have a maximum mass of around 100kg, and linear dimensions that at most are around 3m maximum length, 2m in the next largest dimension, and perhaps 1m at most in the smallest dimension, though most will be considerably smaller. Any bigger, and Santa couldn't lift it to get it out of his old sack. The old guy is probably far more muscular than his suit makes him look, given that he can sling potentially large, heavy presents around with such speed and ease for 24+ hours at a stretch, so something that only a very strong human could grasp and lift alone would seem to be the upper limit to a traditionally deliverable present.
Additionally, while presents in Santa's presence are immune to the extreme speeds and acceleration involved in delivering so many presents in one night by sleigh, simple experience will show that many presents are not immune when away from Santa's presence, as a launched present must be. Some are very fragile, so we may assume a relatively low minimum g-tolerance of about 4g, the maximum that may be expected in a modern airliner.
So... TL,DR: A present has a maximum mass of 100kg, maximum dimensions of 3m × 2m × 1m for a total of 6 cubic meters maximum, and a g-tolerance which may be as low as 4g.
In order to propel a projectile as far as possible with the minimum launch velocity, over a small scale, a launch angle of 45° is optimal, but over a global scale, where the curvature of the world becomes a significant factor, such a launch angle is no longer optimal. Using a software package such as Systems Tool Kit (available as an online or downloaded trial at http://licensing.agi.com/stk/), experimentation shows that the optimal launch angle to achieve a minimum launch velocity sufficient to propel a projectile to the far side of the world is around 30° to the horizontal, and requires a launch velocity on the order of 5000 meters per second.
However, a g-tolerance of 4g is incompatible with any cannon with a practical barrel length that is capable of propelling a projectile to a velocity of up to 5 kilometers per second. How can this problem be overcome? The answer lies with the Gyrojet range of guns: Gyrojet guns appear to be traditional firearms, however they do not fire traditional cartridges, but instead launch miniature rockets. While Jules Verne's Columbiad may have turned any real human crew into paste on firing, rockets have proved to be suitable for use with g-sensitive payloads such as humans. The Gyrojet sysyem launches its miniature rockets at a low velocity and acceleration, and they continue to gain velocity after launch, making it an eminently suitable system, if scaled up appropriately. So, Santa's "Present Cannon" may in fact be more accurately described as a cannon-shaped rocket launcher.
The logistics of packing, loading and firing all these Present Projectiles would be no less impressive than delivering all those presents by sleigh. If we assume a hundred million presents and on average 3 children per household (necessitating only one delivery for all of the children in that household), and Santa's cannon fires from the moment of dusk on the earliest Christmas Eve timezone as the first recipients go to sleep until just before dawn on the latest Christmas Day timezone, a period of around 34 hours, that works out to be 273 deliveries per second. Obviously, then, one launch must contain on the order of 300 deliveries if they are launched one per second with brief pauses to realign the cannon. This means that each launch - if we consider that the average weight and volume of a present may be on the order of 3.3kg and 0.033 cubic metre - must contain a payload on the order of 1000 kg and 10 cubic meters.
I leave it as an exercise for the OP to design a tube-launched rocket capable of lifting 1000 kg and 10 cubic meters of payload at 4g maximum acceleration and propelling it to a velocity of 5 kilometers per second.
So... Santa's "Present Cannon" would in actuality be a relatively thin-walled tube (compared to a real cannon firing similar sized projectiles) with a bore of around 3 to 6 metres diameter, potentially a hundred or more metres long, and with a complex autoloading and aiming mechanism at the base.
For the final part of our design of the projectile, we need to consider the delivery of the presents. Within each rocket would be a delivery pod, somewhat similar in purpose to the multiple independent reentry vehicles of a modern nuclear armed missile, that being to deliver the payload of one rocket to multiple targets. However, presents are not typically g-tolerant, and Santa wants to deliver intact presents, not impact craters. So, the challenge is to deliver the presents at a velocity significantly less than the potential 5 kilometers per second. Fortunately, the relatively shallow launch angle means that the angle of descent will be similarly shallow. While the cannon may be directed far more vertically, and the projectiles pitch over toward their target after leaving the atmosphere, in order to minimise the amount of air it must pass through, a shallow angle of descent increases the distance the projectile passes through the air. This is useful, since the delivery vehicles can use aerobraking in order to shed speed. By using a drag-inducing device such as a ballute, the speed of a supersonic object can be reduced in high atmosphere without requiring an ablative reentry shield. Then, once the delivery vehicle is subsonic, it can deploy a parafoil with which it can maneuver to dock with its target chimney.
Next comes the target: the recipient's chimney. As Santa has traditionally come down the chimney to deliver any presents, even in dwellings which do not normally include such a suitable architectural feature at any time other than Christmas Eve after the residents are asleep for the night, we may safely assume that the magic of Christmas ensures that each dwelling will posses a chimney large enough to allow Santa and the largest possible traditionally deliverable present to descend down it's length. Obviously, this is larger than any chimney found in a dwelling these days, so it forms a standardized target for the delivery vehicle to dock with.
Finally, presents delivered, the virtual chimney disappears, along with the delivery vehicle still attached to it.
Of course, this system would share another problem with nuclear missiles, that of interference between delivery vehicles. This can be minimised by targeting the most distant unserviced target on any given bearing so that subsequent launches result in the projectiles not passing through the descent path of an earlier missile or Delivery vehicle.
