Can you fuel a car while it is running?

Besides it being illegal, you SHOULDN'T for several reasons.

Todays cars use a pressurized fuel system, if the ignition is on while the cap is off it will throw a check engine code due to loss of pressure.

Gas in liquid volume is not as flammable as many people think. You can toss a cigarrette in a 5 gal bucket of gas and it will just go out.

However, gas evaporates VERY quickly and is VERY volatile. You run a greater risk of explosion from holding a cigarrette NEAR a bucket of gas than you do throwing it into the gas. If your gas cap is open, there's a MUCh greater risk of explosion from any errant spark contacting the gas vapours.