Part of a plumber's fee is contained within the markup of his materials. And most will not warrant third party supplied materials. If he installs a faucet and it leaks, he can swap it out and take it back to his supply house to send back to the manufacturer. If he installs your supplied faucet and it has a problem, then you pay him to come back out, and you deal with the hassle of either sending it off to the seller or contacting the manufacturer for replacement parts.

So sure, you can pay a lower price from an online supplier. But, it won't end up being cheaper, and you end up running the risk of it actually being MUCH higher in the end.