There are multiple websites where Connecticut residents can check for a COVID-19 vaccine appointment. The two pharmacy chains, CVS and Walgreens, allow people to make appointments directly through their websites.
The Vaccine Administration Management System, VAMS portal, has appointments listed for municipal clinics, Nuvance, Mall drive thru and others. https://portal.ct.gov/vaccine-portal/COVID-19-Vaccinations-VAMS-Support includes the form to pre-register for access to the system.
Some heath systems though have their own sign up portals.
Yale New Haven Health https://www.ynhhs.org/patient-care/covid-19/vaccine/get-your-covid-vaccine.aspx
Hartford Health Care Network https://hartfordhealthcare.org/health-wellness/covid-vaccine/schedule-your-vaccine
Stamford Health Network https://www.stamfordhealth.org/covid-19-update/covid-19-vaccination-information/
Middlesex Health Network https://middlesexhealth.org/location-finder/middlesex-health-covid-19-vaccine-clinic-by-appointment-only
People looking for appointments through VAMS must first fill out a pre-registration form. An email will be sent to you from the CDC/DPH to create an account. When you log onto VAMS for the first time it will ask if you've registered before and you must click 'no' because the form that was previously filled out was only a pre-registration form confirming eligibility. Then you can search for and book an appointment here https://portal.ct.gov/vaccine-portal/COVID-19-Vaccination-Scheduling-Options.