Reddit reviews Large Deli Cup EZ-Fit Dispenser uses a self-adjusting gasket to handle large paper, plastic and foam bowls and containers used for soups and salads
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Accommodates most 32 to 64 ounce bowls using supplied gasket packsHigh Impact plastic design1 Year WarrantyEZ-Fit Cup Dispensers utilize a patented dispensing system which allows you to easily change cup sizes without having to change dispensers or sacrifice dispensing function. Each EZ-Fit Dispenser includes a gasket pack to handle a large range of sizes.
https://www.amazon.com/EZ-Fit-Dispenser-self-adjusting-plastic-containers/dp/B00165LHL4/ref=sr_1_1?s=home-garden&ie=UTF8&qid=1468091046&sr=1-1&keywords=deli+cup+dispenser
Maybe this could lead you into what you're looking for, although lid storage isn't something I could find. It is what I came across with a quick amazon search (which I'm sure you've done already). I've had luck finding particular items from local restaurant supply stores or cash & carries. Good luck with your search!