I'm not familiar with Zoho, but are you using any other features apart from e-mail? Is this a free service or paid for?
For policy and procedural matters, the decision on whether or not an indvidual allows any inspection of the setup rests with the council, not the individual, so it should be discussed at the next meeting. Confirmation of the ability to view the contents of e-mails should be obtained. Once fully informed, the council can make a decision about whether or not this is acceptable.
In an ideal world, the responsibility for website and e-mail administration should sit with the Clerk or an external service provider, but there are often practical or financial barriers to this and if a councillor has a background in IT, they may be far better equipped in this regard. I manage IT for all three of my councils; two as Clerk and one as Chair.