What sources of weather are approved by the U.S. National Weather Service?
There is an ongoing issue with this, as far as I am aware. First off, this does not apply to part 91, and nothing does as far as I am aware: you can get your weather from anywhere, but having it on record with FSS or DUATs gives you a little more padding if something goes wrong.

If you're going to get weather online, which most large operations do, the FAA requires air carriers to get weather information only from Qualified Internet Communications Providers (QICP). This is outlined under AC 00-62. Or rather, it was outlined there. 00-62 was quietly cancelled in mid 2013. In addition, the only list of QICPs (which would have answered your question) I know/knew about was removed from the FAA website.

A few weeks after it was cancelled I found out about it and called the FSDO to ask (Memphis or Louisville, I don't remember which). The man who should know told me that I knew about as much as he did. Their regional offices hadn't told them anything, and he was telling everyone to continue using the qualified service they were previously using until they learned more. 

I have not called them back since, but after a quick search, I only found this OpSpec document and one that it supersedes from a few months before (8900.199). There is also a newer one in the series (8900.242) but it seems to have no information relevant to the question. I must adimit that this document is a bit over my head, but it seems that they did an overhaul of the weather regulations regarding 135 and 121, and that use of a QICP is no longer required. So I take it that unless your carrier is only certified to use a certain system, you're now allowed to use whatever you want to get the weather.

You can read those documents here, the three I mention were published under AFS-200. Search for that and you'll find them and more.



So I did some more research:

Check out 8900.1 Ch 26 Section 2. Regulatory Sources of Aviation Weather Information and Aviation Weather Information Systems – Parts 91K, 121, and 135. The first bit has a good overview of all the weather products and their intended purposes. Relevance to the question starts at 3-2073(A) Weather Reports Prepared by the NWS or a Source Approved by the NWS. (page 6 in the PDF version).

Much further down it says (in reference to part 135, 121, 125, etc certificate holders only):


  B. Accessing Weather Information via the Public Internet. Certificate holders and 
  program managers often access weather information via the public Internet. When accessing 
  weather information this way, certificate holders and program managers are required to use 
  weather information that is provided by a weather source authorized by regulation, or approved 
  by the Administrator, and is listed in the certificate holders’/program managers’ 
  OpSpecs/MSpecs. In the past, the FAA maintained a list of approved QICP. However, the FAA 
  no longer approves QICPs. The FAA has canceled AC 00-62, Internet Communications of 
  Aviation Weather and NOTAMS, and no longer maintains a list of approved QICPs. Certificate 
  holders and program managers using the public Internet to access weather information are 
  responsible for ensuring accurate and timely delivery of information without data corruption 
  during the transmission.


That's all pretty cut and dry, I think, but keep in mind that those OpSpec papers mention that everything in 8900.1 Ch. 26 is under review and has been edited recently and will be further edited in the near future.