I recently bought and installed a standalone version of Lightroom 5 from Amazon after my free 3 months trial of Creative Cloud had expired. Now I can't seem to use the "Development" function because it's asking me to renew my membership. How do I renew my membership without having to buy a monthly plan via Creative Cloud?
Have you actually purchased Lightroom 5? If you have, then you are still running the creative cloud installation of Lightroom and it will have to be removed. Then you can download and install Lightroom 5.7.1 from this website: Product updates
You first have to uninstall the existing version of Lightroom which is configured for the Creative Cloud. This you have to do using the normal process for your operating system, it will not interfere with your preferences, catalog, previews, or other personal files.
Download the appropriate version for Lightroom and use the serial number you have obtained when you purchased LR 5 from Amazon. The serial number cannot work with the present version you have installed. If you purchased an upgrade license you will likely need the previous version license to complete the activation.
See this link for LR 5.7.1 the last one for LR 5.