Raining, yes, of course. Planes take off and land in rain and even snow. Severe thunderstorms are different and it really depends on the severity of the storm, the wind levels and where the storm is in relation to the airport and flight path. It's certainly possible your flight could be delayed - if the storm is over or close to the airport, they could halt takeoffs temporarily. Be sure to have something to keep yourselves entertained if you have a delay, have extra baby food packed if you have an infant, have a charger for your cell phone in your carry-on NOT checked luggage, etc.
If your flight gets canceled, do not just wait in the sure to be long line for rebooking. Get on the phone with the airline immediately AND get in line - then talk to whichever agent (phone or counter) is available first.