Moving to Vancouver — is ICBC really the only company I can buy car insurance from?

For your basic mandatory coverage, yes — BC runs a public system through ICBC, so that part isn't shoppable the way it is in Ontario. But you can still shop for optional extra coverage (on top of the basic policy) between ICBC itself and private insurers/brokers, so there's still some comparison to do, just a narrower one than you're used to. /guides/car-insurance-by-province lays out how this compares province to province.