Where available, data of the vessel, as well of the owner and his contact details are published in the vessel's datasheet on the bottom of the vessel detail page. All users, independent of their account status have access to the FleetMon data sheets.
However, ship owners and captains hardly ever make the ships contact data available to the public.
As far as we know there is no public database of ship's e-mails or telephone numbers. They keep this information out of the public realm for security reasons and to avoid spamming/advertising and high satellite communication costs for the vessels.
The best method of contacting a ship is through the operator/managing company name which is usually listed in the datasheet on the bottom of the vessel's detail page. Once you have the company name, you'll have to do an old fashioned internet search to get their headquarters contact info.