How do I find out if my municipality offers free/discounted compost?

You call your local municipality office, but often it is the utility company that has the mulch.

I asked about compost not mulch

Compost and mulch are two different things. Compost can be added to the soil for better vegetation and the mulch is usually spread around the base of plants to keep away the weeds. If the compost is well rotted then you can add it now but it would be good to add more in the Spring. As suggested, call your municipality to see what they have.

Thank you!

Em is right. Call the city and ask them.