There appear to be no requirements for the Domincan Republic.
These days, Yellow Fever is about the only vaccination for which evidence might be required when travelling to certain countries.
It is always good practice to carry a note of your vaccinations when travelling, as it will remind you to keep them up to date.
This US site appears to be reliable and up to date.
http://www.mdtravelhealth.com/destinations/mamerica_carib/do minican_republic.html#3