We encountered identical forecasts for rain/storms when visiting Chamonix and Nice in early June. Storms were predicted for every day of our trip. The day before driving from Geneva to Chamonix still predicted storms for the next couple of days.
Fortunately, these forecasts were mostly inaccurate for both locations. Weather patterns in mountains and along the coast are very difficult to predict. In the mountains, it is common for the weather to be clear in the morning, but warm humid air moves in during the afternoon creating clouds and potential for rain. That is exactly what we experienced in Chamonix. Went up the Aiguille du Midi on the 9:55am ride and had mostly clear (not perfect) skies ... absolutely worth it. Periods of light showers (no heavy rains) came in the late afternoon and early evening for an hour or two.
Throw your rain jackets in a backpack and expect you may need to use them. And just grab a seat into one of the many covered outdoor cafes in Chamonix to wait out any rain while enjoying the town's charm. Hopefully the weather will cooperate for you too!