I ran into the same issue (running v13.1.0.2) using Chrome (v66.0.3359). What eventually appears to have corrected it was....
- Use the F5WinInfo.exe utility on the (Window 7) device to uninstall components.
- Connect to VPN again using Chrome and download the f5vpn_setup.exe file then click to install.
- Wait several minutes (I gave it about 10).
- Click the "Click here...when installation completes" link, and the connection will be established, authenticated, and so on.
Prior to that, I tried the Component Installer, but still got the pop-up about client components being required. Also, I wasn't waiting very long for the f5vpn setup file to complete, and since it never gave me any kind of "install complete" pop-up, all I could do was guess.
At this time, Firefox v52.8 and IE v11 are working just fine. Lastly, I made sure the APM clients file was the latest version (see K52547540: Updating BIG-IP Edge Client for the BIG-IP APM system).
Good Luck!