If your Community users cannot access Conga Templates saved as Salesforce Files, please follow these instructions:

You must be logged into Salesforce Lightning Experience to follow these instructions
  1. Open your Conga Template record.

  2. Click Related to view the Notes & Attachments related list where your template file is attached.

  3. Click View All.

  4. Click the Dropdown Arrow to the right of your template file and select Share.

  5. In the Share window expand the Who Can Access section.

  6. Make sure the Customer Access Enabled switch is switched on: