Well, to be pedantic, Wayne and Alfred can do all the work they want on their house by themselves and not alert local zoning authorities. If they get caught they're in trouble, but the only reason they would be caught is after an incident caused the Batcave to be discovered by the authorities. Inspectors don't go knocking on doors to make sure you didn't build a multimillion-dollar secret computer cave in your basement, they'd never actually be able to inspect actual buildings being built as a result.
However, given Wayne's near-infinite amount of financial resources, it seems easier to just have a contractor do the deed for an immense amount of money and never speak of it ever again in exchange for doing all of WayneCo's contracting jobs at no-bid prices. This does result in a number of manual laborers knowing that you have an immense cavern under your house, but the Joker isn't going to be stopping all the constructions workers to chat about such things.