Is there an API to access live (or semi-live?) data from NOAA satellites, such as the GOES-R series? I've been searching for weeks, and I know there are ways to manually access data through website file systems, but I'm looking for something more like, for example, the NASA Mars Rover Photos API, or even just a data stream.
Where are you searching? The NOAA and all of its child departments/organizations provide a plethora of satellite and API datasets, including satellite datasets accessible via API. The list below is a tiny sampling of all three; I cannot stress enough how much more data there is. NOAA is basically a cornucopia of satellite/API datasets.
API Web Service - weather.gov
Web Services API (version 2) Documentation | Climate Data Online (CDO) | National Climatic Data Center (NCDC)
NCEI Data Service API User Documentation
Tides & Currents Web Services
Advanced Data Access Methods | National Centers for Environmental Information (NCEI) formerly known as National Climatic Data Center (NCDC)
NCDC Satellite Data
NCEI Satellite Data Services
NCEI Satellite Oceanography Group
NCDC's ArcGIS REST Services Directory