You probably won’t find it at a home improvement store. When we first did Hamlet, we ordered it from a lumber store. Even lumber stores don’t often carry it. We have done some fixes the past two months (remaking the front kitchen upper face, and both side upper faces) and at the lumber store where we live now, we found 1/8 inch birch ply “door panels.” They were $14 apiece and worked perfectly for what we needed to do. Wouldn’t do if you need a full sheet, but more available. Call around to find someone who does lumber, lumber, lumber. They know their stuff. And you’ll most likely be supporting a local business.
Boulterplywood advertises as a supplier out of Medford, MA. For the Midwest, I picked up 15sheets of birch 1/8 ply last week from owl lumber in Lombardy IL. $20/sheet I see Medford is near Boston. Lombardy is very near Chicago. I called around here, and found most places inventories were variable, even if they advertised as having stock, except the one near a larger city.