Check with someone in your area who is in the business of repairing
musical instruments such as clarinets, flutes, etc.  This is the
business that I am in and we purchase a lot of cork in thicknesses
from 1/64" to 1/4".  We get them in 4" x 12" sheets.  I'd offer the
names and numbers of the suppliers, but most of the suppliers in our
business will only sell to instrument repair shops.

Be prepared to pay a good price. A 4" x 12" x 1/32" sheet of cork will
probably cost in the neighborhood of $10.00 per sheet!  This is, however,
"real" cork; it's not "filled" or changed in any way.

John W. Miller