In the list of social networks in Easy Social Share Buttons for WordPress, you have a subscribe to the mailing list button. That button can show a custom design form (HTML code, shortcode) or you can show an integrated form.
To enable the usage of the button you need to add the Subscribe button to the list of active networks for sharing on your site. There are also a few settings for the design of the form that will be used. Those settings you can find in Social Sharing -> Networks -> Additional Network Options -> Subscribe.
The button has 3 operational modes that can be used.
In this mode, you can place own code (HTML or shortcode) for showing the form. You need to ensure that this code is connected with a service. The plugin will just show that code and nothing more.
Link to Subscribe Page
In this mode, when pressed the button will open the selected URL.
Service Integrated Subscribe Form
In this mode, you can integrate the forms you design inside the Subscribe Forms menu. There are separate fields for inline and pop-up appearance. That is because when you use the button with a content display (like content top or content bottom), the form will show inline below the list of share buttons. If you are using the button in a sidebar display, the form will show as a pop-up.