There is an add on role that gives limited access to a membership record, and the ability to still print, issue, or cancel membership cards directly from a membership record. 

Note: This role does not give the ability to bulk print membership cards. 

Have a System Role Administrator grant the Print a Membership Card role to the user: 
  1. Click Administration > Application Users
  2. Click the arrow beside the username
  3. Click Edit
  4. Mark the checkbox for Print a Membership Card Role
  5. Click Save