Although "having the world at your fingertips" can mean having the world at your command, it can also mean having ready access to the world—in many instances, specifically, through technology or through other sources of information.
A Google Books search finds an example of the phrase used in this sense in the title of an article about short-wave radio that appears in Boys' Life magazine (October 1953):
The World at Your Fingertips
There are few spots you can't reach through the magic of short-wave radio • By KEN BOORD
But the phrase actually goes back several decades before that, with the sense "having everything one might want within reach." Initially, the phrase was a component of longer phrases of the form "the X of the world at [one's] fingertips." For example, from "The Trend of Modern Fiction," in The Advance (April 13, 1905):
One of the most popular fiction writers of the day, Mrs. Humphrey Ward, in her latest novel, introduces as her leading female character a woman who with all the good things of this world at her fingertips leaves her husband and runs away with a poet.
And from a review of John Dillon's Commentaries on the Law of Municipal Corporations in The Yale Law Journal (November 1911):
Quite noticeable also is the large amount of space devoted to the foot-notes. The author has well carried out his purpose of making the book as useful to the village lawyer who is denied access to a good law library, as to the urban counsel with the libraries of the world at his fingertips. This result has not come from a dependence upon unsupported general and abstract statements but from notes which are full enough to be quoted, and from frequent discussion of the leading cases.
But soon enough "the X of the world at [one's] fingertips" gives way to "the world at [one's] fingertips. From Anthony Pryde [Agnes Weekes], Marqueray's Duel (1920):
She had nothing to say to the pleasant subalterns of the Guards' Brigade nor they to her. Yet she was not left lonely, for elder men—Robert Vere, or Mallinson, of the Exchequer, or a smiling Opposition chief, or some shrewd wit from the cross-benches—liked well enough to slip into a neighboring chair. Val had the political world at her fingertips, for Yarborough talked as freely to her as to West. She was as discreet as her father and as effortless in her discretion: her gossip was colored by a warm humanity: she was more interested in other people's lives than in her own.
The more commanding sense of the term as "having unlimited opportunity or power" comes out in this example, from Will Comfort, Lot & Company (1915), where the person speaking is talking about a baby to the baby's mother:
"He is having his adventures. He will be a great man. He will have the world at his fingertips, when he is as old as we are—and then his real work will begin. For when we know enough of the world, we turn to God."
And again, in a paradoxically deferential-to-the-tastes-of-others context, from Theta Phi Alpha, The Compass (1946) [combined snippets]:
Right now, the main fashion prediction is to dress to please our returning menfolks. No more 'jeans and shirt-tails out'—they saw enough of such things where people had to dress that way because they had nothing else. However, the American woman has the world at her fingertips if she will get away from man-tailored clothes and be feminine again.
Christine Ammer, The American Heritage Dictionary of Idioms (1997), offers this definition for the shorter phrase "at one's fingertips":
at one's fingertips Ready at hand, immediately available. This idiom is used both literally, as in This new dashboard design keeps all the important controls at the driver's fingertips, and figuratively, as in Tom was so familiar with the proposal that he had all the details at his fingertips. [Second half of the 1800s]
Although Ammer doesn't note that "having the world at one's fingertips" can a have a different sense—not just of having all things immediately available but of having them under one's control or power—I think it's useful to recall that the "within reach" meaning of the phrase preceded that other meaning.
Ammer revisits the expression to better effect in The Facts on File Dictionary of Clichés, second edition (2006):
at one's fingertips Ready, instantly available; at one's command. The term refers to both cognizance and competence—that is, it can mean either knowledge or the ability to carry out a task. Presumably it is based on something being as close at hand and familiar as one's own fingers. Its roots may lie in an ancient Roman proverb, "To know [X] as well as one's fingers and toes," which in English became one's fingers' ends (in the proverb collections of John Heywood, John Ray, and others). Fingertips appears to have originated in the United States in the nineteenth century.
Nathan Bailey, Dictionarium Britannicum, or a More Compleat Universal Etymological English Dictionary (1736), has this version of the expression:
To have a Thing at one's Finger's Ends. To be very apt at a thing; to know it perfectly well.