I don’t know about AccuWeather specifically, but don’t most weather sites ultimately get their forecasts from the National Weather Service?
Accuweather and basically every forecaster is using input data from NOAA or other government agencies.
The biggest private companies could, theoretically, run proprietary models for their customers if NOAA were to quit doing forecasts altogether. For instance, the GFS is open source so someone with a big enough supercomputer could use it.
Plus, surface readings are only a small part of the atmospheric picture; you need weather balloons to generate a more holistic reflection of the atmosphere at a site. There are also data generated from aircraft, but those are more one-dimensional (in the sense that they are only sampling conditions at a single altitude) than twice-daily balloon launches from selected sites. Until someone decides to implement a private, nation-wide radiosonde network, there probably won’t be a numerical prediction model that operates independently of NOAA data.