Open the Plex search page when pressing enter in the search box.

openPlexSearch.user.js

// ==UserScript==
// @name         Open Plex Serach Page on Enter
// @namespace    https://app.plex.tv
// @version      1.1
// @description  Open the Plex search page when pressing enter in the search box.
// @author       JonnyWong16
// @homepage     https://gist.github.com/JonnyWong16/1ebd38a5f5872408e5b840ed28006dec
// @downloadURL  https://gist.github.com/JonnyWong16/1ebd38a5f5872408e5b840ed28006dec/raw/openPlexSearch.user.js
// @updateURL    https://gist.github.com/JonnyWong16/1ebd38a5f5872408e5b840ed28006dec/raw/openPlexSearch.user.js
// @match        https://app.plex.tv/*
// @grant        none
// ==/UserScript==

window.addEventListener('load', function() {
    const searchInput = document.getElementById('downshift-0-input');
    searchInput.addEventListener('keydown', (e) => {
        if (e.key === 'Enter') {
            e.preventDefault();
            window.location = 'https://app.plex.tv/desktop/#!/search?query=' + encodeURIComponent(searchInput.value);
            searchInput.blur();
        }
    });
}, false);

Did some troubleshooting. If I paste your code from line 15 to line 22 into the console, everything works fine. If I include line 14 and 23 (i.e. the window.addEventListener stuff), it doesn't work.

If I put the code into Tampermonkey, it doesn't work no matter if I include line 14 and 23 or not.

The Tampermonkey icon in Chrome says it is enabled and that your script is active on the Plex page.