I have a website developed by a different program (wix) and I own the web domain to the site. Can I store my Wordpress site on a subdomain of that site? For example my site is website.com - and I want my wordpress site to be located in mysite.website.com .
Is it possible? Absolutely; the how depends on your setup though. It will be slightly different depending on where the site is installed (VPS, shared host, ect.) Without more information I can't give you anymore than just the general steps that need to be taken. That said here they are.
Generally the steps are:
1) Install WordPress on your host.
2) Go to your DNS settings on your domain registrars dashboard and add an A record with the hostname set to the subdomain & address/value set to the IP address of the server WP is installed on. This tells clients which IP address to go to when they look up the domain name.
3) Configure your server software (Apache, Nginx, ect..) to serve up the site whenever mysite.website.com is requested.
You don't have to have the subdomain on the same server as you're main site.