By high speed orbits, you mean they orbit closer to the star than your planet, right?
Getting machinery there is the costlier part, in terms of energy. A lot of delta-V involved. You have to match their orbital speed in order to dock to them.
This involves two problems:
- it is a huge change of speed;
- due to the former problem, it takes a lot of fuel... But in order to take large amounts of fuel anywhere in space, you have to burn more fuel (every bit of fuel is a payload until ou use it).
Essentially your problem is similar to that of catching the Voyager and bringing it back home, as seen on XKCD. Only you want to go to a lower orbit and back, instead of going through a solar escape trajectory and back.
There are ways in which you can solve those problems. Mining done exclusively by drones keeps weights needed to operate low, and it means you can use an acceleration rate that could kill humans. You can also sacrifice your miners with less ethical implications if they are disposable drones.
As for fuel, Nasa is testing an engine that does not require the ejection of mass in order to accelerate in space. That's witchcraft to many scientists, but just goes into showing that you probably don't need to handwave at all to have mining ships of reasonable sizes to reach those asteroids (really, go read that XKCD article and check the figures!).
One last thing. Keep your masses as low as possible. Don't bring the asteroids whole back home. Have the drones separate ore from useless rock. Bring only the ore back home. If your ships launch from the surface of the depleted asteroids, they will also push whatever's left into lower orbits, potentially making the "mining orbits" safer.
To bring the ore to the surface, put it in orbit of your planet first, then have space shuttles take it in chuncks to the planet surface - make it a regular service, just like Space X takes payloads to and from the ISS on a schedule. It then turns into a matter of having enough ships to keep your business productive.