Connecticut's town record storage can pose a challenge if you don't know what town you need. You can spend a lot of time search and come up empty handed.
However, there are a few ways around it.
1) FamilySearch has an online index of Connecticut records. Dates are scattered, but it's a good way to start: https://familysearch.org/search/collection/show#uri=http://hr-search-api:8080/searchapi/search/collection/1674736.
2) For records since 1897, contact the state vital records office. They have a slow turn-around, but they do have an index to every record in the state. It may be worth the extra six weeks.