There is no 'DMCA institution', further issues are handled in US courts assuming that the theme marketplace is operating under US jurisdiction. Read through the DMCA FAQ at http://www.chillingeffects.org/dmca512/faq.cgi paying attention to the notice, takedown, and counter-notice sections. Other countries may have similar mechanisms but details are likely to be quite different.
Assuming that the theme marketplace is in the United States and that your competitor does file a DMCA, you have the opportunity to file a 'counter-notice' with the service provider (theme marketplace) asserting your work is original and that you have the right to distribute it. The service provider is supposed to restore your original work in 14 days unless whoever filed the original DMCA notice brings a lawsuit in court. Whoever files a DMCA notice may be liable for damages incurred by the takedown.
Assuming that the theme marketplace is in the United States and that your competitor does file a DMCA, you have the opportunity to file a 'counter-notice' with the service provider (theme marketplace) asserting your work is original and that you have the right to distribute it. The service provider is supposed to restore your original work in 14 days unless whoever filed the original DMCA notice brings a lawsuit in court. Whoever files a DMCA notice may be liable for damages incurred by the takedown.