Are there any natural supplements or shampoos that can kill lice?

Not really. If you want to kill lice, then you will need a shampoo that has a chemical insecticide in it. It is possible to eradicate lice without using one, but it would require great dedication to check the whole scalp daily for live insects, and comb the hair strand by strand to remove the tiny white eggs (nits). The over-the-counter medicated shampoos are very safe, and i think they are a wiser choice.
Some try vinegar. Some will swear by vinegar, either cider vinegar or pickling vinegar worked into dry hair daily for a week as a natural rx. It does seem to loosen the egg casings from hair but the effect is variable. You could try it & go to a standard pyrethrin rx later if not successful initially.