My business name is registered with the Secretary of State in Arizona. I found someone using "mybusinessname"@gmail.com. Can I insist they stop using it? Can I make them surrender it?
Under trademark law, your competitor cannot confuse your customers to think that they are you, thereby stealing your business and interfering with your relationship with your customers. Here, it is not clear if you are concerned about customer confusion or you simply desire the gmail name for your own purpose.
Any rights you have under your AZ trade name or trademark are related to your common law trademark rights, so, without knowing the how, when, or where of your trademark use and the gmail account use, it is impossible to guess how your rights may help. That is, an Arizona trade name is simply a registration that allows entry into Arizona corporate names database. An Arizona trademark registration does confer additional trademark rights, but, it requires that the common law rights already exist.
I do not suggest sending a demand letter without consulting with an attorney. When you make a legal demand in another state you may become subject to a "declaratory judgment" lawsuit by the recipient of the demand letter.
On the other hand, you should check gmail's terms of acceptable use. This gmail account may be abusing Google's terms of service. Google might find that your trade name certificate is sufficient evidence of a trademark to help you.