Please provide us your site url so that we can take a glance.
Also try switching to standard wordpress theme and look if the customizer works or not.
Besides please deactivate and reactivate the plugins if you have used any one by one and check if the issue still prevails or not. Sometimes third party plugins may create issues. You need to consult respective plugin manager in that case.
Note: Please make a backup of your website before proceeding and be sure you hadn’t have done any customization.