One of the reasons you're unlikely to find an API like this is that not every country experiences those seasons. Basically everything around the equator has a wet and dry season instead of summer, spring, winter or autumn. 

If you want to go for a “simple” version, you could simply retrieve the latitude and longitude of each city from an API and assign a season based on whether they're in the southern or the northern hemisphere, but it will not be realistic for a great number of countries.