The weather-related adjective meaning stormy or tempestuous is inclement. It is the appropriate word in the phrase inclement weather, which means stormy weather. The word is often misspelled inclimate (which would also be pronounced a little differently), but this illogical form is not yet common enough to have received the recognition of dictionaries (at least the dozen or so we checked), and it is not found in edited writing.
The prefix in- is one of several prefixes we attach to adjectives to make their opposites. For example, decent becomes indecent, articulate becomes inarticulate, and credible becomes incredible. But climate is not an adjective and doesn’t have an opposite, so inclimate would have no logical meaning.
The opposite of inclement is the much rarer clement, meaning mild or gentle, which comes from a Latin word with the same meaning. Considered etymologically, then, inclement could mean not mild or not gentle. Incidentally, clement is also the root of clemency, meaning mildness or leniency.
In these sentences, the questionable inclimate would bear replacement with the standard and long-established inclement:
With inclimate weather and Spring Break altering the schedules, many teams, including Tecumseh, had less than five days of practice. [Adrian Daily Telegram]
Wind and other inclimate weather conditions will also make the gym a better venue for the event. [Marion County Record]
[A]t times they received a bad wrap due to bizarrely inclimate weather and an unfortunate piece of judgment regarding a ticketing snafu that impacted a few hundred fans. [Forbes]
And these writers use inclement well:
She climbed Horseshoe Ridge in distinctly inclement conditions. [Telegraph]
We’ll be back to dreary rain in no time, so don’t pack away your inclement weather gear just yet. [Washington Post]
Anticipating inclement weather, GOP officials in Maine’s Washington County postponed their Saturday caucus for a week. [Wall Street Journal]