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 components of 3D WebView that explicitly use the legacy Input Manager are the demo scenes (which use StandaloneInputModule) and native hardware keyboard support (which uses the Input class). So, to use the new Input System in your project, there are several 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".
If you don't want to enable both input systems, another option is to exclude the Demos folder by renaming it to "Assets/Vuplex/WebView/Demos~".
If you want to utilize the demo scenes and only enable the new Input System, then a third option is to update the demo scenes to switch from StandaloneInputModule to InputSystemUIInputModule. You can do that by pressing the "Replace with InputSystemUIInputModule" button on each scene's EventSystem:
Please note that 3D WebView's automatic handling of native hardware keyboard input currently relies on Unity's Legacy Input Manager, so it will be disabled if "Player Settings -> Active Input Handling" is set to "Input System Package".