"Springtime à la Carte" was written in 1906, and in those days they wrote more "literary," complicated English. This is not easy English. And because it was written for entertainment and not as serious literature, it contains cultural references that are not easy, even for native English speakers.
"Gross" means coarse or down-to-earth. Nowadays it has come to mean really repulsive or disgusting. I think O. Henry just means everyday, down-to-earth.
I think the key here is that buckwheat and oysters are things to eat, and are not high-class foods. (Oysters are expensive now, but they were cheap, ordinary food in New York in 1906).
I think he is making fun of flowery writing. You'll notice that he makes fun of writers and writing in several other places--he opens the story "It was a day in March. Never, never begin a story this way when you write one."
He is talking about "harbingers of Spring," things that tell us that spring is on the way. First, he talks about nice, pretty, sweet, refined, conventional signs of spring: "the flowering crocus, the wood-starring dogwood, the voice of bluebird."
He then talks of "the farewell handshake of the retiring buckwheat and oyster." Oysters were available in months whose names contain the letter R. The first months of spring, March and April, would have been the end of the oyster season. I don't know about buckwheat, I'm guessing it is the same. "Spring is the season of the year when we have to say goodbye to buckwheat and oysters." A modern equivalent might be "the Christmas season, when we have to say farewell to Pumpkin Spice lattes." Notice, too, that the food reference sort of foreshadows other food-related elements in the story.
In other words: artistic, lovely, sweet, refined flowers and birds, versus "gross," plain old everyday, low-class food.