If it's not feasible at all to send a projectile into orbit like this then we'd like to know if it's feasible to perhaps grab the projectile at the height of it's trajectory.
Something a little like this may be feasible. The basic idea is to launch the projectile from the highest altitude you can manage, on a fairly flat trajectory a little faster than orbital velocity. The "little faster" is chosen so that, when the projectile leaves the thicker part of the atmosphere it is in an elliptical orbit with apogee where you want it, and perigee somewhere close to the launch altitude. There is a bit of a tradeoff here -- launch more vertically and you leave the atmosphere sooner, more horizontally and you end up with a higher perigee.
Now you need to intercept the projectile roughly 45 minutes later at the apogee of it's first orbit and accelerate it enough to raise the perigee of its orbit clear of the atmosphere (that may be as little as 100 m/s).
You can raise the perigee a bit (at the cost of some velocity) by designing the projectile to get some lift from the atmosphere as it rises. You can use a fairly small rocket to circularize the orbit instead of catching the projectile if you want to.
Now, let's see about this cannon. You need the projectile to leave the cannon at (ballpark) 8-9 km/s. This immediately rules out chemical propellants, and pushes the limits of a hot gas gun. Your other options are basically electromagnetic launch of some kind, or a nuclear explosion. 100kg is a really too small for the nuclear option which seems better designed for 1000 ton plus payloads, so we are probably left with an EM launcher. This is definitely possible. The length depends mainly on the acceleration your payload can stand. For example a 1km launcher would mean about 3200g ($v^2 - u^2 = 2as$) and the acceleration would take about $250 ms$. The energy you need to deliver to the payload is ($1/2 mv^2$) about $3.2 GJ$ so the power delivered is about $12.8 GW$, plus extra for inefficiencies (but this is for a short time, so could come from capacitor banks). The force on the projectile is $320 kN$. Other lengths of accelerator will give you other numbers.
The accelerator will basically be an evacuated tube (which could be made of fibreglass or similar) surrounded by magnetic coils. The smaller you can make it the easier it will be to manage the magnetic fields, so if your aim is just to launch 100kg slugs of metal or something, then you probably want to make them long and thing (also helping with air resistance). So the barrel might only be 10-30cm wide. You have to manage the vacuum seal at the exit end and the shock waves when the projectile hits the atmosphere. This may need a widening of the barrel at the end (and possibly some reinforcement). The seal could be a thin membrane that is just destroyed by the slug coming through it, or it could be opened rapidly as the projectile approaches (some care needed) by hydraulics or more magnets (or even a small conventional explosion).
