Do you need property insurance even though your business is online?

Depends - what do you sell online? If a council, and you can afford to replace your computer and other business equipment, probably not.

However,

If you sell a product, you want coverage if the burning of values? If so, is likely to have property insurance. And you need a liability insurance anyway, if you want another person (the insurer) to pay for your defense if you are sued.