Access permission for Print Merge Documents
Also, if you are willing to develop an extension, you need to check https://docs.civicrm.org/dev/en/latest/hooks/hook_civicrm_summaryActions/ hook to add permission to the "Print/Merge Document" action link.

Something like -

function mymodulename_civicrm_summaryActions(&amp;$actions, $contactID) {
  $actions[print_pdf_key_name]['permissions'] = array(...); //your list of permission
}


If you need to create a new permission for this task - you need to check https://docs.civicrm.org/dev/en/latest/hooks/hook_civicrm_permission/ hook as well.

Hope that helps.