One potential emergency, for which having a good three-month supply would be invaluable, is a flu pandemic. Health & Human Service Secretary, Michael Leavitt recommended that every person store food in preparation for a possible pandemic.
"When you go to the store and buy three cans of tuna fish, buy a fourth and put it under the bed. When you go to the store to buy some milk, pick up a box of powdered milk, put it under the bed. When you do that for a period of four to six months, you are going to have a couple of weeks of food. And that's what we're talking about." (Preparing for a Pandemic)
That all seems like a lesser version of a three-month supply to me. In fact the church's link (below) discusses sheltering-in-place in a pandemic for up to 12 months. With a three-month supply you would be able to shelter-in-place for the duration of any quarantines.
For additional information in preparing for a pandemic, this link (Pandemic Preparedness Planning) connects to an obscure corner of the Provident Living Website. There they discuss the following pandemic preparedness topics: home and family preparedness, personal hygiene, personal protective equipment, infectious disease cleanup, sheltering in place, social distancing, workplace preparedness, and health care worker preparedness.
The scenario of a pandemic is worth discussing with your family. And having a three-month supply would allow you to stay out of public stores and better protect your family from any infectious disease emergency.