There are some obvious advantages to using supercaps in buck converter designs including low output voltage impedance and high output currents. But those same advantages also effect the circuit design.
Ideally, with a supercap, the charging happens on a time scale such that it would be possible to just connect the input voltage to the cap, monitor the cap voltage, and then disconnect the input voltage at the desired cap voltage. However, in practice, this method draws high currents from the input requiring an expensive high watt power supply and beefy cabling or traces. So, putting an inductor and a transistor switch in series with the input is desired, just as they would be used in a standard buck converter. But in the case of a supercap buck, the inductor is there just to limit the input current to levels that are tolerable for all of the components. The power throughput of the buck, then, is dominated by the inductor selection.
One should choose a current rating for the transistor switch and shunt reverse diode to be greater than the input power supply current rating, and the current rating of the inductor should be greater than twice the input power supply rating, then select an inductance and PWM switching frequency such that at a 50% duty cycle, the integral for the first half (when the switch is “on”) of a PWM period of the inductor current divided by a whole PWM period reaches 80% of the power supply max continuous current (more current will flow through the inductor after the transistor is switched off, but that current will come from the reverse diode rather than the input power). This process often requires a few iterations as both the available PWM frequencies and available inductances are often restricted. When calculating inductor current it is safest to assume that the cap voltage is always set to zero to account for the possibility of a very low impedance load on the secondary. The PWM output to the transistor switch gate is, of course, enabled/disabled by voltage feedback to the controller, which will limit the cap voltage to the desired level.
For the capacitor, I suggest the use of Licap 350F or 3000F cells to insure maximum power delivery to your coil gun, I mean… to your safe and beneficial application.