i heard chicks cant shampoo down there because that stuff dribbling into your vag could cause infection
I don't think that's the case. If it were they'd never be able to wash themselves at all with shampoo or soap since it all rinses down the body and around there anyway.
Girls aren't supposed to wash out their genitals normally though, as you do with a douche, since that can upset the bacterial and pH balance down there, which can allow for infection, such as thrush.
The female genitals are designed to not allow liquids and things in constantly, to prevent this. Hence why washing normally is fine. When you attempt to wash the inside of the genitals then it can cause problems.