The livechat button is rendered on the website page when the livechat module is installed and configured properly. The livechat module uses JavaScript to load the button on the website page, and it can be placed anywhere on the website as per the configuration.

Cookies play a role in determining whether the livechat button is rendered or not. The livechat module uses cookies to keep track of whether a visitor is a new visitor or a returning visitor. If the visitor is a new visitor, the livechat button is rendered on the website page. If the visitor is a returning visitor and has already interacted with the livechat, the livechat button may not be rendered. The livechat module uses cookies to keep track of the visitor's interactions with the livechat, such as if the visitor has closed the livechat or if the visitor has already initiated a livechat session.

It is also possible that the livechat button is not rendered due to a technical issue, such as a JavaScript error or a problem with the livechat module configuration. If the livechat button is not rendering as expected, you may want to check the JavaScript console in your browser for any error messages, and also check the livechat module configuration to ensure that it is set up correctly.

Additionally, you can also check the livechat module's settings, it might have the option to show the livechat button only during specific hours or days, or you can check if the livechat agent is online or not.

If you are still facing the issue, you may consider checking the logs in the Odoo server and also you can reach out to the Odoo support for further assistance.