Yes. Use the scripts workspace (Manage | scripts in FMP 13 and older) to create a script with the send mail script step. You can use the same dialog to set up the recipients address, subject, body, etc. but now select the "perform without dialog" option. Now you can set your button to perform this script to send your email.
Note: sometimes issues with the email client program may still require some user interaction. This depends on the version of FileMaker, the email client program and version plus how the email client program is configured.
Thank you for your message.
You can do a number of things using the "Send Mail" script step. You can hardcode the content of your email directly into the "Send Mail" script step if you'd like OR you can base the content and addresses of your email from data you have in your solution. And if you want it to automatically send without prompting, you can use the "Perform without dialog" option that is part of the "Send Mail" script step.
You can get more information on the "Send Mail" script step here.