To see a list of all events in the system, select Monitor > Event Console from the Console Launcher.
To see a list of events associated with a selected inventory object and its child objects, perform the following actions:
1. Open the vCenter Management console.
2. Select an inventory object.
3. Select the Monitor tab.
4. Click Events.
At least I was not able to locate any filtering facility in web client. Though it mentions on pg.108 that it is available there.
How to export logs from vSphere Web Client
Switch console views to the Event Console view and click Export.