Our site is penalized. We asked Google to remove it and they sent us a bunch of links that caused them. All these links were sites with wordpress theme footer links that trace back to a site offering free wordpress themes. We did not set them up, and we know our competitor did this. We told Google webspam team about this and they don't care.
What happens now? I never realized this, but Google is supporting the uprising of the negative SEO industry. Any company can harm someone else by creating free themes and templates and putting their links on the bottom footer.
Does this mean the only way forward is just to run negative SEO campaigns against my competitor?
Several site owners has similar issues. But the issue that I am having with concern to the problem you stated is that How come one site has WP theme links, I mean you must have one theme and so the link could be directly from that. However, if Google is listing it then it might be considering it as unnatural. Try clearing links apart from the WP theme one and then resubmit your site for checking, make your statement properly as to what you did and what you left along with the reason. I am sure they will help, its just that because of too many request the response can be delayed.
If it still shows then I think you should use the disavow tool.