Earth is literally the only place in the universe where accurate astrological readings can be made.
(I'm not into astrology, so I'm sure I'll get some of these details wrong.)
First, the star constellations that make up the signs of the zodiac are all defined with respect to an observer on Earth. While it's true that the constellations will look essentially the same from Earth's moon, and probably even from other bodies in the solar system, they will look very different from any other place in the universe. And if you're already going to the trouble of traveling back to the Sol system to take readings, you might as well travel the last few AU to visit Earth.
Second, the zodiacal year is also defined with respect to an observer on Earth:
The zodiac is the belt or band of constellations through which the Sun, Moon, and planets move on their journey across the sky... twelve of the constellations through which the sun passes throughout the year
If you really want an accurate reading of the current zodiacal moment, the best way is to bring your kit to Earth and take a look.
Now, you may be thinking that a person could save themselves the effort by just using software that accurately simulates the view from Earth. Software like that exists today (astronomers use it). But I'd remind you that astrology is a pseudo-science (i.e. a crock of ****), and we can probably assume that any future adherents who care enough about getting a "true" reading are the same kind of people who are unlikely to put their trust in a piece of man-made software.
And, in defense believers: everyone must acknowledge that a simulation is only as good as its data and models, so if there really were some kind of mystical truth in astrology, it seems quite possible a simulation based exclusively on hard observation would be unable to anticipate a novel occurrence. And there's Humean skepticism: the recognition that we have no good reason to believe the future will be like the past. These people will not trust a mere app.
So, I'd expect permanent astrological observatories on the cold Earth, who sell their observations to humans across the galaxy. I'd also expect there to be some very wealthy lunatics who insist on traveling to Earth to make their own observations. And possibly some people who are unable or unwilling to pay the fee charged by the observatory, who therefore have no alternative but to visit Earth to take their own measurements.
Just to be clear: when I talk about readings and measurement, I am not talking about writing horoscopes or making predictions for individual people. That stuff is built on top of the readings. The readings are just this: stand on Earth and look into the sky to observe the apparent positions of the sun, moon, and constellations. That is how you determine the "current zodiac time."
Folks who want their horoscope would pay the Earth-based astrologers to tell them that info as it was at the moment of their birth, which is how the person determines their "sign." (A person would presumably only need to do this once per lifetime.) And then they can go and get a horoscope from their preferred local astrologer.