If your administrator has specified it, Anyconnect has a Diagnostics and Reporting tool (DART) module inlcuded in its installation. You would see it as a "Diagnostics..." button after clicking the gear icon in the AnyConnect user interface. (see below)
Also, the VPN logs are kept in a Windows application event log. Look in your Windows Event Viewer > Applications and Services Log > Cisco > Cisco Anyconnect Secure Mobility Client.