Cursor locking (using Cursor.lockState) and full-screen mode (using Screen.fullScreen) are both supported in Unity WebGL, implemented using the respective HTML5 APIs (Element.requestPointerLock and Element.requestFullscreen). These are supported in Firefox and Chrome. Safari cannot currently use full-screen and cursor locking.
Due to security concerns, browsers will only allow locking the cursor or going into full-screen mode in direct response to a user-initiated event (like a mouse click or key press). Unfortunately, Unity does not have separate event and rendering loops, so it defers event handling to a point where the browser no longer acknowledges a full-screen or cursor lock request issued from Unity scripting as a direct response to the event which triggered it. As a result, Unity triggers the request on the next user-initiated event, rather than the event that triggered the cursor lock or full-scree request.
To make this work with acceptable results, you should trigger cursor locking or full-screen requests on mouse/key down events, instead of mouse/key up events. This ensures that when the request is deferred to the next user-initiated event, it is triggered when the user releases the mouse or key.
Note that browsers may show a notification message or ask the user for permission before entering full-screen mode or locking the cursor.