Could you provide W3 Total Cache settings or WPEngine(which has inbuilt cache mechanism) or .htaccess caching, I should add in order to make the plugin work?

Follow below mentioned steps:

1) You need to add "Request a Quote" page and admin-ajax.php to exclusions list of the caching.

2) Exclude cookie generated by woocommerce from caching starts with "wp_woocommerce_session_". Eg: wp_woocommerce_session_79d7ec763f2d6998b91c4289e4f4e738

It varies site to site. If you want I can help you to find out for your site. If you know little knowledge on how to see cookie info in the browser, you can do yourself.

3) May be you are caching the pages in .htacess file. Please check this too.