We have a number of farmer markets in my area with all of them being year round. Saturdays I can walk across the street to a small farmers market for salad and micro greens, Indian food ingredients, artisanal items, a vegetable breakfast crepe and of course chocolate hazelnut truffles. On Wednesday there is a very large farmers market with a huge selection a few blocks from where I work and on Sundays there is one in size that’s between those two and closer than the Wed one. Mostly on Sunday and sometimes on Wed I get berries/fruit, veggies, Maggie Farms organic greens/herbs, Asian ingredients, great breads and on Sunday sometimes breakfast with friend/s. Heirloom tomatoes, ‘Orange Flesh’ or smaller ‘Sun kiss” melons and Flame and Black Emerald grapes are all in the Sunday and Wed. markets at the moment. I can ride with a melon or hard fruits/root veggies in a day backpack and other foods just in a front basket (with a thick anti-vibration foam pad in the bottom) with harder foods on the bottom and berries, eggs, greens on top. Never had a problem with any of it. I use a soft canvas grocery bag that fits in the front basket. Crossing the handle straps over the bag and attaching them to the sides of the basket keep things secure. We have a market refrigerator in the office for people needing to use it on Wed or store those market foods for lunches.
At both the Sunday and Wed markets there are bike racks set out for around 75 or so bikes and they are usually close to being filled during market hours.

For eggs I just keep reusing a container I got with store bought eggs that keeps them reasonably cushioned and protected. Never had a problem.