Create a vest to detect contact
The idea I'd investigate for doing it is to have a conductive vest which uses principles of capacitance, where the bullet would impart a charge stored on it to the vest. This could make for extremely cheap vests and relatively cheap bullets; also they would be very light and robust. I expect the issue to overcome would be the best way of storing charge on the bullets. But using FET transistors which are very sensitive to small charges would mean the bullets may need minimal static charge. Although how then to protect the FETs from damaging static electricity may be a bit of a dilemma. It may even be possible for the bullets simply to become charged merely by passing through the air.

A magnetic system might be doable if you have something like arrays of small coils on the vest, which when the magnet approaches at speed, would induce a current/voltage. The problem is the weight and cost of the vests could be prohibitive. Also the vests might be delicate.

Using light could be a problem, as with the magnetic system, because unless the bullet emits a large amount of light, you would need a lot of sensors on the vest. That is, unless you can find a material that directs light from around the vest to a couple of sensors. Something perhaps along the lines of a number of fibre optic cables, or a material with an equivalent property.

An extension to the first idea, perhaps have bullets that behave a bit like RFID tags. A conductive vest could be used to energise the bullet, via radio waves or similar; While the bullet is near the vest, a coil and/or tuned circuit in the bullet is energised by the frequency of the vest. The vest then detects a change sent back by the bullet, or perhaps the modulation effect of the bullet on the vest's own oscillation. This idea definitely needs significant refinement, as it is way beyond my experience.