It does depend on locale. In US cities where there's some density the postal worker will usually leave their truck (which is often a regular, non-custom, mid-size box truck) at the end of a street, fill up a wheeled bag, and walk to several buildings with the bag to drop things off, before returning to the truck and moving to the next location.

But in most suburbs (which is a lot of the US), at least where there are mailboxes at the curb (and not mail slots on houses), the postal worker will just drive from mailbox to mailbox, dropping mail as they go, never leaving the vehicle, only getting out of their seat when they need to add more mail to the box sitting next to them.