Operating a 0.08 amp fan using capacitor
The required energy, in joules, to power the fan for 10 seconds can be calculated:

9.6 joules = 12 volts * 0.08 amperes * 10 seconds

The required capacitance needed to store such amount of energy in a capacitor:

0.133 farads = 2 * 9.6 joules / sqr(12 volts)

The capacitance needed is equivalent to 133 millifarad or 133,000 microfarad and the closest E10 value is 150 mF or 150,000 uF. Using a radial aluminum electrolytic capacitor capable of handling 16V, the capacitor physical dimension is 18mm in diameter by 36mm in length.

To answer your question: Yes, it's possible to use a capacitor to store energy for powering the fan.  A capacitor solution is simple as you just need to connect it across the load (fan) and no complicated charging circuit is required. However, your may be constrained by the following:


high capacitance capacitor requires a large space
high capacitance also means a longer charge time: the fan may not be powered for the full 10 seconds if power is lost while the capacitor is still charging
a big capacitor charging will cause a high inrush current at power on: your supply must be designed to handle that
a capacitor will not provide a constant voltage as it is discharging: you may need to use a higher capacitance so the full 10 seconds discharge will be within an acceptable voltage range for the fan
big capacitors are not easily available: may require a special order
a battery module may be cheaper than a big capacitor


I leave to you the evaluation of the cost-benefit tradeoff between a capacitor and a battery.