Do the DVR from these companies have built-in tuners?
Yes. They all have tuners which will receive their own HD programming. But only the satellite companies' tuners include an over-the-air tuner (in case you also want to use it to get your local OTA's with an antenna).
You can also purchase standalone tuners which can receive the local HD/SD digitals through cable (QAM tuner) or OTA (ATSC tuner).