WooCommerce can be either a plugin or full WP theme, it's still served up on web pages regardless which one you choose. Web page vs web page, sure, you can compete against another site. Whether you rank your page is up to you & what you do with your own web page/s & what you can do as far as SEO.

I've built a site using Woocommerce and it's pretty straight forward for the most part. Best of all it is a plugin like yukon said, so you can perform all the proper onpage SEO changes you normally with (install Yoast or Light SEO plugin).

Only thing is you might want to use a made for Woocommerce theme (have some free ones like Mystile), otherwise there could be some compatibility issues with CSS, etc.

Backlinks, keywords, content, social sharing etc... those are the things that get you traffic, not so much your platform. Customizing your woocommerce theme with SEO in mind can help to some degree, mostly just to make your keywords stand out.

The basic goal of good SEO is just to make sure you get relevant customers... the more the better, but for long term gains, they need to be relevant.