I would use it, but I would store the fabric in expensive cotton pillow covers.
Acid-free tissue paper and boxes are pretty expensive.
I store fabric in hard plastic boxes, but I would hesitate to store it for long periods of time in soft clear plastic bags, as some of them outgas chemicals that eat fabric (plus it's easier for moisture to remain in them, and trapped moisture would cause mold in the fabric -- which is impossible to get out later).
I really think cotton pillowcases are the safest and least expensive way to go. Alternatively, you could make a cloth lining for the cedar chest.
The big thing is that you do not want to store fabric in direct contact with wood. The acids in wood (and in wood products such as cardboard, paper) will gradually eat into the fabric and cause it to stain and deteriorate.