Can I sign up at one site and get approved as a mortgage broker with several lenders?

In theory, you are not committed to any lender until you lock an interest rate. But brokers don't get you approved at multiple lenders... Simply doesn't work that way. They may work with multiple lenders... and have options to send your file to multiple lenders, but you are "not approved" per se with multiple lenders. Rather, your loan officer will run your file through automated underwriting to get a basic approval - which is good with basically everyone.

You will need to contact and qualify with each lender you represent.

Sounds like you are asking as a mortgage broker and if you are I unfortunately don't know of any one source to get approved with that will approve you with several lenders. That would be a good idea though.