Having just moved some accounts over to Vanguard recently, there are several ways to do it. I elected to have my former brokerage liquidate everything, as they would do it without fees, and then have Vanguard pull the money over. They just did it directly, not by mailing you a check. I believe this is preferable for tax-advantaged accounts, to avoid accidentally doing a withdrawal with tax/penalty consequences. This resulted in some time out of the market, about a week or so.
You can also elect to have Vanguard do an in-kind transfer - basically pull over the assets in the account over directly. So you owned X shares of Fund A before, now you still have them, just in Vanguard. Then you would have to direct Vanguard to sell those shares (probably incurring some sales fee) and buy new shares. The time out of the market would be however long that sell-then-buy transaction takes - maybe a day or two? If your old brokerage charges transaction fees too, I would probably go with that option.
Edit to add: This can be done on the phone as mentioned later, but you can frequently do the whole thing online. I believe you get at it via the My Accounts -> Open Account button, but that's from memory.