Hornbeams are widely used for hedging so should be cheaply and widely available.

If planted as a specimen, rahter htan densely in a hedgerow, they are pretty fast growing. It will be 6-9ft fairly quickly even if you put it in smaller one, and give it some food and water. If they are being anal about it being 6-9 ft when planted, you could buy a small one and grow it on in a container until it is 6-9ft and then plant it, but it will establish better if planted smaller.

Who should plant it, why you. Dig a hole and chuck it in, that's about it really. Do it during the dormant season, and make sure the roots are well spread, untangle them if potbound. You can get them bare-rooted in winter.

Councils are inclined to make ridiculous restrictions about plants you must plant. For example my parents were required to plant a hedgerow along a fence line as negotiated agreement for some planning thing. They tried to specify the exact composition of the hedge and also specify a specific contractor should plant it (probably the planning inspector's brother), but such are unreasonable requirements, and they were able to insist that they should be able to plant it themselves, of small plants, and a less restrictive description of the species mix: they didn't want a high proportion of hawthorn, nor any blackthorn, for example, in their garden.

Is there any reason they have specified hornbeam? If there's something else you'd prefer, why not fight for it? Hornbeam's is native and common as muck around here (often mistaken for beech so frequently overlooked), no showy flowers, whereas a horse-chestnut is a non-native decorative specimen with showy flowers. Why not replace it with another decorative specimen if that is what you want. I don't see why they should tell you exactly what.

I would have thought if they learned another way it was unsafe, they'd be at your door instructing you to cut it down. But when you tell them, they make you beg to be allowed to and instruct you what to replace it with, cheeky wotsits.