This was tested on Red Hat 7, but should work on most Linux systems. For Debian variants you will have to use a different package manager and the package names may vary.
1) Install the necessary packages.
yum -y install coolkey pcsc-lite*
2) Start the pscsd service/socket
systemctl status pcscd.service
3) Open Firefox and add coolkey module to security devices
Go to Preferences > Advanced then click the certificates tab
Click "Security Devices"
Module Name: DoD PKI
Module Filename: /usr/lib64/pkcs11/libcoolkeypk11.so
4) Install DoD root certificates
Scroll to bottom of page under " PKI CA Certificate Bundles: PKCS#7"
Download " For DoD PKI Only - Version 5.0"
Extract the zip file and inside the uncompressed directory you will find 3 certificate files named similar to:
5) Go back to Firefox, Preferences > Advanced and Click Certificates.
Click View Certificates to open the Certificate Manager
Click Import at the bottom of the screen and import the 3 files mentioned in step 4. You will have to do one at a time.
That's it. You CAC should now work without issue on Firefox.