Can't help with logistics companies, sorry.

But for 'safari' Kruger NP has some facilities designed for big groups. Skukuza has a conference centre, that the Skukuza Safari Lodge was built to complement, has over 100 rooms. Most rest camps can also help arrange weddings, catering, etc. So the Safari Lodge or Skukuza Hospitality Manager would be a good place to begin (https://www.sanparks.org/parks/kruger/contact/camps). I don't know if they have a preferred supplier for game drives (I doubt Sanparks would have enough guides and vehicles but you can arrange with outside OSV operators) . This assumes you are far enough ahead to block book most of the hotel and a chunk of the vehicle fleet of a larger OSV firm (Kurt, Elephant Herd, etc). Ground transfers from and to Johannesburg would be reasonably easy to arrange and an agent would deal with that. Similarly Nkambeni run by Aha on the boundary of KNP has 150 rooms and catering by buffet, designed for large groups. There are probably several other large resorts just outside Kruger NP who can arrange accommodation, catering and guided drives.

I am sure there will be other options as well, it is only ~20 vehicles, a fraction of gate quotas at KNP. But you would need to booking well ahead (over a year?) to get one accommodation option. So depending where the sports happens there may well be other options, e.g. Pilanesberg, KZN or Eastern Cape.