I'm planning to take an elderly couple from Shanghai to Nantong. I'm looking for a more comfortable alternative than the intercity buses.
Are there any companies who offer intercity taxi or minivan for a decent price?
Well if you flag down a regular Shanghai taxi they would either refuse because it is out of their way or be very happy about a huge fare (but my guess is the former).
This kind of operations in China are typically not a company with a website but rather a guy who makes his money driving people around in his van, and if it were you'd pay a big price premium. I suggest that you ask the reception of your hotel in Shanghai or Nantong to arrange for a driver (I suggest to also agree on a price beforehand), they should be able to do this kind of thing.