Carol Hein, 72, a retired teacher from Prescott, Ariz., on her 1918 Buick E-35, as told to A.J. Baime.

This year our family Buick turns 100 years old. I have an idea to call the folks at our local service station and tell them I am bringing in an ’18 Buick for an oil change, then take it down there just to see the look on their faces.

The...