I'm using com.atlassian.stash.mail.MailMessage and com.atlassian.stash.mail.MailService to send an email, but I can't configure it to send out HTML.

In MailServerImpl (line 324) message.setText(mailMessage.getText()); is being used, but that doesn't provide a possibility to flag for HTML content. Therefor you need to use helper.setText(..., true);

And of course there is no possibility in the MailMessage to state the content should be HTML