Truthfully, I pick them when I have time, when I notice that they are ready, or when I need them. Thus that could be anytime during the day. But I do have a habit of working the gardens early in the morning right after tending the livestock. So it's common for me to be harvesting between 8 and 9. But if I'm in the process of making dinner at 6 pm, I surely wouldn't hesitate to go pick veggies and herbs.
Do I notice a difference in the flavor of things picked in the early morning versus evening? No, not really. But then I'm no food connoisseur.
I've read recommendations that harvesting for best quality should be done in the early morning. The reasoning is that overnight the plants have had the opportunity to rehydrate (absorb water) and convert the starches produced the previous day into sugars, thus the fruits and veggies would be more crisp, juicy, and tastier. I notice that most veggies are definitely crisper in the early cooler mornings.
One thing I learned, regardless of when the veggies and fruits are picked, it's best to rinse them in cool water and keep them cool. That rule doesn't apply to everything, but most things stay fresher if kept cool and moist. Things that I don't cool down are potatoes, sweet potatoes, pumpkins, and bananas. But I do get them out of the sun.
Some veggies I'll store in the kitchen in a vase. Vase? Yes. Celery, Okinawan spinach cuttings, sprigs of various herbs, cut greens (broccoli leaves, spinach, etc), Chinese cabbage, bok choy. I'll use a vase when refrigerator space is in short supply. Sticking veggie stems in water will keep them fresh for a couple days, especially useful if I happened to pick too much or changed my mind about what to make for dinner.