No. The WebViewer component links to the Android system WebView, not one of the browsers installed on the Android. WebView is not a full featured browser. Android WebView is a system component powered by Chrome that allows Android apps to display web content. This component is pre-installed on your device and WebView links to it. Google updates the version of the system WebView from time to time.
My understanding, WebView inherently has no storage. To provided storage, code needs to be written in the WebViewer component and might be possible.
In all cases, the WebViewer component uses the system's native WebView implementation. On versions of Android prior to 5.0, this was a custom WebKit implementation bundled by Android. Starting on Android 5.0 the system can use another engine, and on most Android devices approved by Google this is Google Chrome. The WebView interacts with Chrome through a separate background process and renders the page in the view's bounds. As the user upgrades Chrome on their phone, the WebView gains functionality.
However, some functionality needs additional setup from the WebView's client app. We don't currently enable every feature of the WebView, though. There are some extensions that do so you might want to consider those for your project if you really need localStorage. In theory, you can also use the WebViewString and WebViewStringChange blocks to implement a custom storage solution, but then the page contents will be specific to App Inventor.