Jan Dembowski on "Wordpress and ssl"

Is it possible to use relative URLs in wordpress without the use of a plugin?

You could but you'll hurt yourself doing it. All of the coding for WordPress is designed for absolute URLs and forcing relative URLs will break things such as RSS feeds.

I am trying to implemet ssl with woocommerce and it is prouving rather challenging so far.

If your site is ready for TLS, why not just implement the Site URL and WordPress Address settings as https: URLs instead? If you also set HTTP Strict Transport Security (HSTS) then your visitors will only hit your site via an encrypted URL.

https://en.wikipedia.org/wiki/Strict_Transport_Security

That is normally set on your web server level though it is just an http header so you should be able to set it via a plugin or your theme's functions.php file.

Another reason for using relative URLs is security related

No, it's not.

*Drinks coffee*

I'm sorry, "security" is a thing for me. ;)

There's no more security in relative URLs than there is in hiding your URL from users. Which you do not want to do. The security is in the code and how you setup your site. If you are conceed about it (and that's good) then give this a read.

http://codex.wordpress.org/Hardening_WordPress