3D WebView's prefabs detect input through Unity's event system like described here, so they automatically work with input modules that send input through Unity's event system, including the new Input System's InputSystemUIInputModule. The only component of 3D WebView that currently still relies on the legacy Input Manager is its support for detecting input from the native hardware keyboard. So, to use the new Input System in your project, there are two options:
The first option is to enable both the new Input System and the legacy Input Manager by setting "Active Input Handling" to "Both" in "Player Settings" -> "Other Settings". This is the recommended approach, as it allows 3D WebView's logic that detects native hardware keyboard input to still function.
If you want to enable only the new Input System, you can do so by setting "Active Input Handling" to "Input System Package" rather than "Both". However, this will disable 3D WebView's logic that detects native hardware keyboard input, since it currently relies on the legacy Input Manager like described above. In other words, 3D WebView will not be able to automatically detect when you type on your PC's hardware keyboard. If you choose this option, your application can still manually detect and handle hardware keyboard input.