Important note to begin with: A page can't query if an extension is installed without explicit help from the extension. This is done to prevent browser fingerprinting and/or preventing sites from denying content if certain extensions are installed.
WebExtensions are largely built upon the same principles as Chrome extensions. As such, this question is relevant: Check whether user has a Chrome extension installed.
However, some of the best methods available in Chrome are currently unavailable in Firefox:
The files will then be available using a URL like:
moz-extension://<random-UUID>/<path/to/resource>
This UUID is randomly generated for every browser instance and is not your extension's ID. This prevents websites from fingerprinting the extensions a user has installed.
As such, what are your options? The page can't talk directly to the extension context (background), and the background can't directly affect the page; you need a Content script to interact with the page content.
How can page code and a content script communicate? They are isolated from each other unless content script does something about it.
First off, generic tricks that work in both FF and Chrome:
- You can create or modify a DOM element on the page from a content script and look for those modifications in the page. -   // Content script
  let beacon = document.createElement("div");
  beacon.classname = browser.runtime.id;
  document.body.appendChild(beacon);
  // Page script
  // Make sure this runs after the extension code
  if (document.getElementsByClassName("expected-extension-id").length) {
    // Installed
  } else {
    // Not installed
  }
 
- You can use - postMessageto communicate between contexts, though it's clunky to use as a bidirectional channel.
 - Here's documentation and sample WebExtension. -   // Content script code
  window.postMessage({
    direction: "from-content-script",
    message: "Message from extension"
  }, "*");
  // Page code
  window.addEventListener("message", function(event) {
    if (event.source == window &&
        event.data.direction &&
        event.data.direction == "from-content-script") {
      // Assume extension is now installed
    }
  });
 
- You can use custom DOM events in a similar way. 
There are interesting Firefox-specific approaches as well:
- You can share code with the page using - exportFunctionor- cloneInto:
 -   // Content script
  function usefulFunction() {
    /* ... */
  }
  const extensionInterface = {
    usefulFunction
  }
  window.wrappedJSObject.extensionInterface = 
    cloneInto(extensionInterface, window, {cloneFunctions: true});
  // Page code
  if (typeof window.extensionInterface !== "undefined") {
    // Installed
    window.extensionInterface.usefulFunction();
  } else {
    // Not installed
  }