While major airports may use an AWOS (Automated Weather Observation Station) to collect weather conditions, only the ATIS (Automated Terminal Information Service) is broadcast over an aviation frequency at those airport (at least when speaking of the US airports, I cannot speak with any authority for those outside the US). This is due to the fact that the ATIS contains much more information about the airport besides the weather, including runway closures and conditions, taxiway closures, pilot weather reports, active SIGMETS (severe weather broadcasts), frequencies to use, etc. Thus, AWOS stations at major airports are not able to be received via frequency. Note: Most AWOS/ASOS-equipped airports do have local phone numbers that you can call, however. If you want a US airport AWOS phone number, let me know and I will look through the Airport/Facilities Directory to post it for you.
As you know, the issue with ATIS broadcasts is that the broadcast is pretty much is a continual stream of the same information looped throughout the hour (there may be a 30 second break at ATIS change, but otherwise it is continual). Thus, a volunteer feeder here would have to tie up a sound-card port and a scanner to receive to what amounts to a minute's worth of information repeated over and over. Since looped information would tend to lose the typical listener pretty quickly, just about all of the volunteer feeders here probably do not have the resources to devote to it.
That said, it appears that there are a couple of feeds that might offer it:
KBDR-Bridgeport CT/KHPN/ZBW - Jason, who is very active in these forums, mentions in his feed detail that the ATIS is available upon request, so you might want to ask him.
SKBO - Bogotá, Colombia (#3) - list an ATIS, but that feed looks to be down.