No, you don't need large capacitors.
From the wiki of Ignition coil https://en.wikipedia.org/wiki/Ignition_coil
So a taser powered by car battery and ignition coil can do dozens of high voltage hits. Hitting a normal human with the full power will likely kill him on the spot, but we can imagine that the agent controls the output with some knob.
As for the delivery method, I assume it is the same for the normal taser: a bunch of wires with arrow heads are launched at the target and make contact. An arc lightning will hit the closest uninsulated surface - usually the agent holding this contraption unless he wears protective gear. In the later case it will arc to the ground.
So yeah if you want to be realistic the Taser Cannon is way too underpowered (either in power or in number of hits per battery) for what we could do with the 90's technology.
I am not complaining actually, hitting that Black Lotus Avatar with it was very satisfying.
The reason why the clip has to be capacitors is not the voltage.
It is that you can't possibly empty a chemical battery in 10s.
A chemical battery is composed from electrochemical cells. Each cell has two electrodes and an electrolyte which reacts with them.
This reaction pushes electrons into one of the electrodes and pulls them from the other.
Speed of any chemical reaction depends on:
Temperature, concentration of products, concentration of reactants, and surface. (and obviously what reaction it is.)
To get the fastest reaction at any given temperature, you have to have only reactants, in the perfect ratio, perfectly mixed.
This is not the case with batteries. You have the electrodes and the electrolyte is between them. Concentration of reactants is how charged the battery is. Concentration of products is inverse of that, combined with resistance, impedance and capacitance of the circuit.
Voltage of a battery is how far it is from equilibrium of the reaction. (Increasing resistance slows down the reaction, because the produced electrons can't leave that quickly, so it can't pull them on the other side as quickly either)
So even at 0 resistance, the reaction will still be slowed down by the electrodes not being mixed in, and by the battery not being perfectly charged.
Not that you would want to discharge a battery in 10s. It is an exothermic reaction. Waste heat would light your gun on fire, vaporise the electrolyte and cause the battery to explode.