Q:

Could a sounding rocket or a space gun payload at apogee be put into orbit with a rocket impulse from there?

Consider a sounding rocket that launches its payload to several hundreds of kilometers of altitude, straight up. Imagine that this payload at apogee then points itself accurately and has a propulsion system that accelerates it by about 8 km/s to enter and stay in orbit.
Would it make sense to launch to orbit that way? Since it has never been done AFAIK, what are the drawbacks with it? What fraction of the mass at apogee would have to be devoted to the propulsion?
Since Israel launches its payloads to the west against Earth's rotation, for reasons of international politics, would it make sense for them to rather launch straight up as a sounding rocket before orbital insertion? And would it in general make inland space ports more attractive for range safety reasons?
If orbital insertion at apogee fails, wouldn't it be relatively easy to recover the parachuted payload for another launch attempt? As in no heat shield required. Compared with a payload misplaced in orbit with today's common kinds of launch failures.
Now, consider the payload of a space gun (or Verne gun or Newton gun to honor the concept inventors) that is launched straight up without any orbital speed component. Suppose it has a solid rocket engine, or some other non-fragile propulsion method (surviving the violent gun shot acceleration, as the payload of for example fuel), that then accelerates it to about 8 km/s required to enter and remain in orbit. And the gun could launch many small payloads frequently at cheap electricity bill costs. Would that make sense in competition with conventional chemical launchers?

A:

The orbital velocity for Low Earth Orbit is around 7.8 km/s. Accelerating to that speed requires that the payload of your sounding rocket or space gun be a rocket that itself has a Delta-v budget of (logically) 7.8 km/s. The Delta-v of an entire launch that gets the same result is 9.4 km/s. In other words, the vertical component of a rocket launch is actually a very small overall portion of the required acceleration. Now, That doesn't actually mean that the rocket would shrink by a proportionally small amount since the "first" delta-v is more "expensive" in various ways. So I'll take a real example.
I'm going to borrow some calculations that I'm assuming are correct which I obtained here regarding the Delta-V that the Falcon 9 has in each of its stage at its stated maximum payload

Start with a 22.8t payload.
At launch the rocket masses 606.8t, at burnout 171.8t.  So the delta V is 300*9.8*ln(606.8/171.8 ) = 3710 m/s.  (The 9.8 converts ISP to m/s).   Assuming the fairing jettisons at stage separation (not quite true), the second stage starts at 138.8t and ends at 27.3t.  So the delta v is 348*9.8*ln(138.8/27.3 ) = 5546 m/s.  Total delta-V is 9256 m/s.

(emphasis mine)
In other words, the upper stage of the Falcon 9 has a mass of about 140 tons total, of which about 20 tons are the final payload. So, assuming the ratios don't change with size (I suspect that's not true, but I don't know for sure) only 1/7 of your space gun's payload mass would actually be available... And even then it's not sufficient since that design only gets you 5.5 km/s not 8. You would need either huge advancements in rocket efficiency or an even worse payload ratio to actually get to orbit that way. By the formula used in that quote, and assuming that the ~4.5 tons between stage 2 mass after burn (27.3) and payload (22.8) accounts for components like the rocket engine and fuel tanks that will stay the same, you actually only get about a 10 ton final payload into LEO that way. So, using (approximately) the most efficient second stage design, only 1/14 of your space gun's mass-to-touching-space capacity gets to orbit.
Whether or not it's a good idea really depends on cost, and what you need to get into space. If your space gun can launch 400 kg at a cost that makes sense to put a 30 kg satellite into orbit then that's fine and all, but it won't really compete with the rockets that put 20 tons in orbit unless it can fire 300 ton upper stages.
Note that I only addressed the space-gun specifically. That's because, as Giskard42 mentioned in a comment, launching an upper stage into space on top of a rocket is just a two stage rocket. Doing so with the first stage on a sounding trajectory is just less efficient. 

