Order from VitalChek
You can also order a copy of a Connecticut vital records certificate from the VitalChek website, which is a nationwide commercial website run by Lexis-Nexis, or by calling them to place an order over the phone. They even offer the option of overnight shipping. However, VitalChek adds on a substantial shipping and processing fee to all vital records orders, not just the overnighted ones. And after placing your order, you may still need to fax them your "tangible interest document" or notarized statement before they will process your order. They are a third-party commercial vendor, not a state or local government agency.
If you are ordering a record from VitalCheck, whether online or over the phone, only credit cards are accepted.
Note that Connecticut "wallet size" (short form) birth records and Connecticut civil union records are not offered through VitalChek's website.
VitalChek also does not have a method to allow members of genealogy organizations authorized to do business in Connecticut to have access to birth records less than one hundred years old, so if this is the legal method by which you are trying to gain access to a more recent birth record, you should apply for the record through the state or locality instead.