You can't e-mail participants directly from within WodUp. However, you can easily export all your participants name and e-mail addresses.
Click on the Athletes link in the competition navigation:
You can then click on Export and save all the competitors to e.g. Google sheets. You could then upload this to your mailchimp list, or e-mail the athletes from gmail.
If you'd like to contact participants of any of your past competitions, and not just a specific one, then click All Participants in the main navigation:
From here, you can export the athletes as above.