How might the story be different than if it had actually been written in 1909? First, specifically think about what things I would have to do to make the setting correct- what specific kinds of things would I need to consider. What things would I need to include or not include. Second, consider what might give away the fact that my story was not actually written in 1910. Se(Hint- if you are a Star Trek or Star Wars fan, consider how the different iterations of those shows changed based on when they were filmed.)
These things may have been mentioned in part, but the first thing that comes to mind is the vast difference of the setting:
Shopping would have been a very different experience. The store would have been much smaller, and someone behind the counter might well have gathered purchases for a customer rather than having the customer serve him- or herself.
In terms of characterization, young ladies would probably have been middle-class, and entered a store in the company of a chaperone or their mother. Young men might have helped as clerks in the store to serve the women (predominately) who visited. The only young girl who might come into the store would be a servant, and she would need to be quiet, waiting for others to be served before her; the upper-classes would have sent servants rather than coming themselves.
Money might not have exchanged hands, but things might well have been charged to a house account that would be settled later, much like buying on credit today.
There probably would have been no confrontation; a young man like Sammy's character would not have made a scene. Had he been unhappy with the behavior of his boss, he would probably have said nothing. Being rebellious would cost one his job, and at the turn of the century, a young man working would have indicated a need for financial help, probably at home, for the upper-middle, and upper-classes would have had no need for their sons to work.
Things that you might want to consider in writing a story set in that time period would be modes of transportation, what kinds of writing implements were used (I think pens with their own ink had been invented by then), how lighting was provided, communication, the social structure, and even watches.
I'm not sure if the others have metioned this, but you would have to be very mindful of how living in modern circumstances have influenced your perceptions and how your present your ideas as a writer. We now have more knowledge about domestic violece and women's rights, but in older times, women were considered men's property and they needed to be "kept in line."
You would have to do an immense amount of research to be sure you didn't slip up and slip in something which hadn't been invented yet, or that you didn't have your characters use words or slang that weren't in use (this just came up in my son's Christmas play when the modern an traditional ornaments were taught a lesson by Joseph--the star and snowman were putting down a new ornament from Hawaii, telling her she didn't belong. One of the kids in the play said that Joseph should tell them to "put a sock in it" and the director laughingly responded that Joseph wouldn't have said that since he probably never owned any socks...not to mention that slang phrase meaning "hush up" wouldn't have been in use in Joseph's time).
Setting is the biggest element, to be sure, not to mention the effect it would have on dialogue, vocabulary and slang of the time, social customs and ways of interacting with others that would have been different in those days, and the kinds of inventions and technologies that could or could not be part of the story.
As the previous poster states, the writer would have to change the story's setting completely as well as the context. Rather than have the girls enter the A&P in bathing suits, why not have the girls enter some other establishment--perhaps, some place where women do not usually enter. Set the story in Prohibition times and put the girls in a SpeakEasy or some similar situation in which all will at first stare at ther.
I think the biggest difference that you would have to think about is how to convey the shocking and radical action of the girls in entering in bathing suits but contextualised to the time you have selected. It would have been unthinkable for the girls to do in 1910 what they did in the original story. You would also have to adjust the voice of the narrator as well - his way of speaking, tone of voice and use of expressions is perfectly suited to his time - but not to the time you are thinking about. Just two ideas, but really everything would have to be reworked. One way of thinking about it would be to start by thinking of the themes or message that the author is trying to convey and then start from there, "re-clothing" the other elements.
Japi, setting has a huge effect on a story and its elements - everything from characteristics to mood to language and register. Let's say, for example, that you were writing a horror story. Horror stories today are full of technology, high-tech tools and gadgets, cell phones, television sets, electric saws, video cameras, PA systems... you see where I'm going with this. A setting over a century old likely would not have contained these things and would require some plot adaptation.
Next, you would have to consider the social standards and roles of the era. For example, women in 1910 were still largely submissive to men and accepted their roles as the weaker sex - the feminism movement didn't even begin until later that decade! Women still were pressured to wear dresses and not have jobs and be married with children by a certain age. This would affect your characterization and the roles of your characters.
Economically speaking, buildings would be made differently, materials would cost less. It would be highly unlikely that your characters, unless very wealthy, would even have a car; transportation would be affected.
On top of all that, any allusions you would make to cinema, music, television, or history would have to cater to the year in which the story takes place.
Taking a year of origin or occurrence into consideration impacts every part of story writing. Unless, of course, one of your characters has a time machine. :)