Yes, cmercury web push supports both HTTPS & HTTP.
Login to cmercury platform and click the website dropdown menu from the web push module and add your website. While you add your website, select the HTTP/HTTPS option depending on whether your website is HTTP or HTTPS. Place the pixel generated in your website header either directly or via Google Tag Manager. Additionally for HTTPS websites, ensure to upload the cmercury SDK files to the top-level root of your site directory, making them publicly accessible.