Solar system data API?
Here is a short answer. @barrycarter's helpful comment links to a much larger list of resources. Also, you may find answers below [this question] helpful. 

There are Spice kernels for many (but not all) exploratory spacecraft as well, you can view them through the Horizons interface or find them via ftp. 

For most objects in Earth orbit, Two Line Elements (TLEs) from Celestrak or Space-Trakck are useful. See this answer as well.