The short answer is no. While highly recommended, it's not mandatory. The extent you need an accountant (or any other professional service provider) will depend on your type of business, revenue, structure.
For example, if you're running a hobby business out of your garage while employed full-time, you might simply need a bookkeeper.
If however, you intend to build a company servicing other businesses or the general public, I strongly suggest you seek a good team of advisors immediately. It will be an investment well spent.
If in doubt, err on the side of caution. Talk to an expert and discuss the business model before starting, if only to find out what you do and don't need.
Jef Lippiatt Owner at Startup Chucktown
I don't necessarily know if you need one from a legal aspect, but having one you can trust is always a good thing. However, if you don't currently have an accountant in mind, put yourself in a good position by using software (Quickbooks, Xero, etc.). Doing that will help set you up for success when you transition to having a bookkeeper or an accountant and they will appreciate that as well.