Would it be possible for a society to start an industrial revolution based around electric engines and windmills?
It might be possible, but with an non avoidable delay.
Thermal engines are much simpler than electrical ones.
First (workable) steam engine is ~1775, with theoretical work (Carnot) in 1824+
First workable electric generator (Pacinotti "ring") is 1860 with theoretical work (Maxwell works) just a few years after.
Wind power has been tapped into many years ago (Holland and Greek mills are centuries older), but they are confined to certain geographical areas (due to wind variation); this is not a (severe) limiting factor because also coal is not that widespread (this concentrated industrial revolution in certain areas like England and Germany).
OTOH the severely limiting factor is transport: while coal (or gasoline) is easy to transport in small quantities this is not true for electrical power; this means a wind/electricity powered Industrial Revolution would be completely different from the coal/steam one; the former would rely much more om large static plants than the latter.
We used fossil fuels because they were there and they were easier than the alternatives - not because there were no alternatives.
As AlexP said, water mills powered the first factories.