The answer to your question is 'Yes', but the Captain does the ceremony on their Private Island,
Castaway Cay.
DCL has booking options for vow renewals and weddings at the Rates and Dates area on their website.
After reading the package details, it says an 'Officiant' to perform the ceremony.
Here's a link to the DCL Webpage that discusses their Weddings at Sea package:
Disney Wedding at Sea
It said it costs $2938 per couple.
I haven't done this, just read about the package earlier.
By the Way, Congratulations!
hope that helps