Technically, no, ShareASale doesn't strictly require you to have your own website to become an affiliate. They do, however, require a live, content-rich online presence according to their FAQ https://www.shareasale.com/info/affiliates/
This means you have two options:
Website: This is the preferred method. Your website should have good, relevant content and avoid inappropriate material.
Social Media Page: You can promote products through established social media accounts like Facebook or Instagram, but these pages should be relevant to the products you want to promote.
While a website might be the easier path to approval, ShareASale acknowledges success can be achieved with other methods. It's important to remember that promoting products without a website might require more effort or investment to build a following.