I attend a church in West Hartford, CT. We are in desperate need to attain fabric for our mission opportunities and a question regarding who may be a provider for tax deductible options as well.
Put a want ad on Freecycle and Craigslist.
Go to fabric stores in the area.
Go to estate sales and garage sales.
You do not say what your mission project is or what size/type of material you are needing but I feel sure if you visit places in your area (and state your mission project) you will be able to find what you need.
There are several thrift stores in your area and many of these may have fabric they will donate or sell very cheap.
Contact (in person is always better) church thrift stores, Hospice Thrift stores, Habitat Thrift Store. Use your zip code with thrift stores.
Place an ad on your local Craigslist (very easy and free) in their free section asking for donations but be sure to give a little more information. hartford.craigslist.org/
Be sure to place an ad in your local Freecycle (with more information). groups.freecycle.org/
Contact your local craft stores - Jo Ann's, Michael's, etc. - also, since this is an ongoing project you might want to contact some of the sewing supply stores as some of these give sewing classes which would mean there are people there with supplies.
For our prayer shawl ministry I got names and addresses of yarn companies. I wrote to several. Some didn't send anything but several sent me odd colors and dye lots not same numbers.
You could also connect fabric stores PR and see if they want to contribute.