I'm going to answer this question based on my own area and experiences.
Bug: June – A lot of bugs seem to come out during June. Beetles called "June bugs" are a thing, dragonflies start flying everywhere, and everywhere you walk, you hear cicadas making a ton of noise.
Dark: August – Crimes tend to be high in August.
Dragon: June, July, August – Dragon-types are based on reptiles and reptiles tend to come out when it's very hot.
Electric: May – May seems to have the most thunderstorms by far.
Fairy: February – Valentine's Day and fairies seem to go hand-in-hand, at least to me.
Fire: June and July – Hottest months of the year, definitely. We had some major heat waves during the these months this year in my area.
Flying: September – Birds tend to migrate a lot during this time.
Ghost: October – Halloween, no question.
Grass: March and September – Springtime arrives, the grass gets greener, the leaves on the deciduous trees return, and flowers bloom. In September, the leaves fall from the trees and the environment is often covered in leaves.
Ground: August – Earthquake activity seems to occur the most in August.
Ice: December, January, February – Arguably the coldest months of the year in many places.
Normal: August – The only month (at least in my area) with no major holidays, which sounds pretty normal to me. I can't really speak for other locations and cultures though.
Poison: November – It is said that pollution is at its peak in November.
Water: April and May – It rains a lot in April and May, at least in my area.
I couldn't really think of anything for the following types:
Fighting
Psychic
Rock
Steel