I saw a message board post recently which made me cringe. The original poster planned on order in a record through VitalCheck. A second poster suggested that she save some money by ordering it through the statewide office of vital records - provided she didn't mind waiting for six to eight weeks. All of these options have their value, but the original poster knew the person's name, the place, and the date the event occurred. She should have started somewhere else.
In Connecticut, if you know where the event occurred, <em>ALWAYS</em> start with the town or city. Depending on the location, records may be in the town clerk's or health department. They'll need verification of your relationship to issue birth certificates less than 100 years old, but all other records should be issued upon request. The town will charge you $20 a record (which is a statewide standard). They can often overnight the records, provided you pay for the service. If not, most have a turn-around of a week or so. Call them and ask for details before requesting.
If you don't know the date of the event, then turn to the state office. They have an index to all events in the state beginning in 1897 (when duplicates of the records were required). They do have a much slower turn-around time of 6 to 8 weeks, so you may want to consider checking area towns first. Most towns will confirm is if they do or do not have the record. If you need the record overnighted from the state, then consider VitalCheck... And if you've used it, I'd be curious to know how it works. I've never had to try.