The user profile page has a link to View and Print Membership Card. But, for that to work, you need to set up the page with the appropriate short-code:
See short-code description and options here: All Short-codes
When the user clicks the PRINT link in the upper right side of the page they will be directed to a printable version of the page:
The featured image used for the page will be printed so do not set a featured image unless you want it printed.
You can also copy the spms-membership-card/templates/membership-card.php file to your active theme’s directory and edit it to change the card template.