Can a potato battery charge a capacitor?
Yes, that should work.   Use a large value electrolytic capacitor, like 2200uF or 3300uF (voltage common value 16V, but any voltage would work.)  Let it charge for half a minute.

You'll need at least two taters wired in series, and use a red LED, since those only need about 1.7V or so.   (Green and blue LEDs need much higher forward voltage.)