I have done a lot of elfa closets from The Container Store, and dealt with some odd spaces. You might have to think outside the box for this one. Something like a wire book rack, or wall racks of baskets.

In the end, if you do have to do wood shelves. perhaps place a cookie rack on top, under the linens, or replace the door with a louvered door. or, be sure to just open and close the closet door every day!

I was browing the Container Store, and saw the wire dish shelves you can get for kitchen cabinets (they are sold everywhere, Target, Walmart, hardware stores) I’m not sure how wide your closet is, but there must be one of those long enough to either be stacked from the floor up (and secured here and there to the walls), or attached to the walls with brackets or clips, or even put in a couple of wood shelves, and wire dish shelf on top of each. so kind of every other shelf is a ventilated one.

Or more outside-the-box thinking:

Click here —>Wine rack!

Maybe look under storage ideas for sports gear? for tools and hardware?

Think outside the box: what you're looking for might not be sold for home closets. For instance, there are vented steel server racks that are 8" or 9" deep that might be repurposed, but they're expensive. Of you could get a local handyman to create vents in wood shelves you've bought that will fit.

Ikea makes a shelving unit that is 9.875" deep, if you can squeeze out the extra inch. Amazon sells 8" deep metal wire racks.