The Swashbuckler's Rakish Audacity method of using Sneak Attack requires that you be adjacent to your target
The feature states (emphasis mine):
[...] you don't need advantage on the attack roll to use your Sneak Attack against a creature if you are within 5 feet of it, no other creatures are within 5 feet of you, and you don't have disadvantage on the attack roll. All the other rules for Sneak Attack still apply to you.
This means that all three of those conditions must be met to use this new method of Sneak Attack, while also following all the regular Sneak Attack rules. One of those three conditions is that you must be within 5 feet of your target.
An overly long sidenote on a different printing
Note that the Sword Coast Adventurer's Guide (SCAG) has the feature worded as follows:
[...] you don't need advantage on your attack roll to use your Sneak Attack if no creature other than your target is within 5 feet of you. All the other rules for the Sneak Attack class feature still apply to you.
This might have ramifications for whether this works at a range since the Xanathar's version (the one quoted first) clearly answers this while the SCAG version has some ambiguity. Does "no creature other than your target is within 5 feet of you" imply that your target actually is within 5 feet of you?
I would say that it does, the same way I would say "Nobody other than Bob came to session" implies (if not just means) that Bob did come to session. Thus "no creature other than X is Y" allows us to conclude that X must also be Y and then that your target must also actually be within 5 feet of you and thus the Sneak Attack cannot be done at range even with the SCAG wording of the feature.