If my company website does not have my business address because I am not open to the public, do I have to use an address?

With respect to whether your company must disclose a physical business address, if your company is a corporation or a limited liability company (meaning registered as such with the Secretary of State's office), then is has to comply with the laws, rules, regulations and other requirements administered by the Secretary of State's office of the state in which the company is registered.  For example, the Connecticut Secretary of State's office requires that a registered corporation or LLC must provide a physical address (P.O. Box is not acceptable), and it must designate an "agent for service of process" which is a person who agrees to receive legal documents that have to be served on the company in case someone sues the company.  That agent is also required to disclose a physical address; a post office box is not sufficient.  This information pertains to the requirements in filings with the Secretary of State's office for a company that is registered as a corporation or an LLC.   You did not state whether your company is registered as such.  If it is not a corporation or an LLC, then the requirements of the Secretary of State that I described are not relevant to your situation.

You asked specifically whether the company must state an address on its web site.  There is no legal requirement that a company do so.  On a web site is not necessary to mention any type of address.  Therefore, your company can do business entirely by internet, without mentioning an address, if that is what you prefer.