Imagine that you want to buy a home. You might find a real-estate agent to show you around, which is a very 20th-century way of doing things. Or you might go 21st century and use the Web to research prices and available properties and to take a few virtual tours.

When it comes time to buy, however, you will probably revert to procedures that were created in your grandparents’ era. You will assemble financial documents and present them to a loan officer at a bank, who will take weeks to determine what you can borrow and at...