I don't know if and how you can create a button like that.
But: You can configure the popup of the layer so that it shows a clickable Mail link. That works for Layers in a web map /app (through Portal). It doesn't work in Pro for me, but that might be restrictions of my institution, maybe it works for you...
Configure the popup of your layer (in Pro or in Map Viewer), switch to the HTML source.
<a href="mailto:{MailField}">Send me a mail!</a>
<br>
<a href="mailto:{MailField}?subject=Parsed%20subject%20line">Send me a mail with a pre-written subject line!</a>
<br>
<a href="mailto:{MailField}?subject=Parsed%20subject%20line&body=Parsed%20body%20content">Send me a mail with pre-written subject line and body content!</a>
Have a great day!
Johannes