Is there a relation between Buddhism and Astrology? Is Astrology present in all Buddhist traditions? Did the Buddha or his disciples make use of Astrology?
In terms of horoscopes, it is definitely not a Buddhist practice to explain one's future or decisions based on some external forces (planets).
However, phases of the Moon have some importance in Buddhism. Many significant events in Buddha's life coincided with the Full Moon - birth, giving first teachings or death, to name a few. Full Moon is also considered to be the Amitabha Buddha Day (Buddha of Limitless Light and the principal Buddha in Pure Land Buddhism). One can find full lunar calendars where different Moon phases are associated with various Buddhas, practices etc. One example of observance days based on the lunar calendar is Uposatha when practitioners intensify their practice and make conscious effort to keep the 5 precepts.
One may wonder why bother with the Moon. In the simplest terms, the reasoning behind it is that our body is mostly composed of water and just like the oceans, the body (with its energy channels) does react to the Moon phases.
The main point is that the Moon phases are not used to predict one's future but rather to mark favourable or auspicious days for specific practises. One could coordinate one's practise with the Moon so that in different days one could focus on specific Buddha forms or teachings. So the Moon should be considered as a support and not an oracle of some sort. It might be helpful but it is not necessary to follow it.
As far as I know, Buddhism doesn't have an overlap on Astrology.
In astrology, it is said that the planets and stars can affect on us and our future can be predicted according to that. So they also provide some remedies to survive from disasters as well. As for their explanation, astrology is an extended version of mathematics which uses the placements of planets to predict about people's behaviors.
But in Buddhism, it is said that everything that happens is based on what is being previously done/happened. So the people make their own fate from the things they do and the choices they make. It may be hardly depending on some planets and stars far far away. But it never says that astrology is false.
Personally, I have seen incidents where astrological predictions became true. And many cases where it haven't. But Buddhism, it is true. It's only a matter of fact how far you're gonna believe and understand it.
Hope this helped.
I come from a Japanese Zen background and currently practice in a Engaged Buddhist sangha under Thich Nhat Hanh. I have never heard of a connection with Zen Buddhism and astrology. That is not to say there is none though. And even if there is no historic connection between the two, if one is mindful they can see the dharma in all things. I do not have knowledge in astrology, but I would bet that if one looked at astrology with mindfulness that you could find the dharma there as well. May you be happy.