Hi Marc.
Based on the information you shared, your competitor has a small but clear advantage over you with respect to links. The answer most SEOs would provide to your question is focus on a link campaign.
I take a different approach which has been very successful. When I begin working with a site, my 100% focus is the on page factors starting with the home page. You have not shared your site URL nor that of your competitors so I can only offer general information.
For purposes of this example let's say you are selling Vitamin Z (mythical product). I would work to ensure your site was the best Vitamin Z site that exists, i.e. make it a world-class site. How?
1. Trust factors. Look at the bottom of the vzw.com site. I would establish a trust symbol block which included McAfee or Verisign, TRUSTe, BBB, Verified by Visa, MasterCard SecureCode along with any industry related symbols. For Vitamin Z an industry leading natural product seal would be helpful.
2. Social engagement. If your site has a login feature, then ensure your login is integrated to allow users to join with their accounts from facebook, twitter, google+, etc.
3. More social engagement. Prominently display (i.e. not in the footer) social sharing options from fb, twitter and google+ at a minimum. These sharing widgets should appear next to any article or product.
4. Allow User Generated Content. Web 2.0 is all about creating conversations with visitors. Allow visitors to create content by accepting comments were appropriate on articles. You can look to integrate a forums, or allow user created articles as well.
5. Expert content. Engage specialists, doctors and other experts for your industry. Ask for their feedback, their assistance in educating the public, etc. You can write the article and ask experts to review it. WebMD often uses this approach.
6. Review your navigation. This critical step is often overlooked. Your navigation should smoothly and logically present your site's content. Determine how easy it is to locate content. Ask others who are not familiar with your site for feedback. Can they immediately locate the information? Or do they stumble around trying to find it?
7. Provide a search function. This feature usually requires a database or CMS. Users like this feature, along with an HTML sitemap.
8. Provide a clear Call-To-Action. When a visitor lands on your site, how clear is your message? Is there a large, bright red ORDER NOW button? Does your CTA button compete for attention with other messages?
9. Add video content. It may be expensive to produce, but it is very marketable and can share your message in a way that words cannot. It also opens up YouTube and other channels for your site.
10. Integrate current events into your site. Using the Vitamin Z example, anytime Vitamin Z is mentioned in any news article either directly or indirectly, you should include that content on your site. Ensure your site is the world's number one source of all information on Vitamin Z.
11. Crisp, clear web design. We live in a world strongly influenced by appearance. Looks count. Some sites pay thousands for web design, and it's worth it! Create a design that is so smooth and great looking that web design sites link to it as an example of a great looking site.
The initial focus of this work is the site's home page. Once completed, it is relatively easy to integrate the changes throughout the site. There are other steps to take, and we can talk about meta tags and site linking, but the initial SEO focus I recommend is begin at the beginning, your site's home page. All your remaining work builds upon the solid foundation you create.