The conductors supplying the plates would also be at a million volts, so would also need insulating with something of huge dielectric strength. You also need to consider the discharge characteristics of a capacitor - almost the inverse of a battery, so the energy would need to be exploited in a fundamentally different way.

It's about how the vaccine is taken up by the body. IM is a much slower release than IV which minimises reactions, especially anaphylactic shock and more serious side effects.

Vaccines are usually delivered via the intramuscular route. It's all about how the vaccine is taken up by the body. Training focusses on the upper arm and thigh, though the upper arm is by far the most convenient. There are other benefits - there are comparatively few nerves in the upper arm muscle, which is why people often feel very little. Those that refute that, well you just got unlucky and the vaccinator hit a nerve. If the vaccinator hits the bone (usually a lack of muscle tissue) you would not know.

The exception is some people (eg young children) may have vaccines delivered through the nose as a spray or drops. This is also linked to efficacy of delivery.

There are always exceptions, but delivering a vaccine other than the intramuscular route would not be a licenced use of the product. So yes, there are legal reasons too!