This question has come up a few times in real life, not so much online. Alchemy has definitely played an important role in the founding and growth of science, the tricky bit is explaining how and why, because it isn’t always obvious. Especially since alchemy was completely written off by the early scientists in their attempts to establish the scientific method and their way of doing things. This meant that until the early 20th century or so, historians and scientists didn’t take alchemy seriously and apart from a general agreement that it was the forerunner of chemistry, it was assumed to not have been much use in the growth of science. Fortunately we know differently, thanks to the last century or so of research.
To make it easier to understand, there are two categories of contribution to the rise of science that alchemy made.
First, there is equipment and techniques.
Perhaps the biggest and most important is distillation; it was perhaps known of before alchemists, but they (in conjunction with physicians) used and expanded it and used it to spread knowledge of distilled spirits of wine etc, and through their experimenting discovered mineral acids, made by distillation.
The mineral acids are useful in testing and producing pure gold and silver and of course the concepts of acids and alkalis was to play some part in the formation of chemistry as a separate discipline.
Alchemists in general were responsible for using and passing on all the general techniques of chemistry, from solution, calcination ,distillation, cohobation, coagulation et, without which the chymists and early chemists would have had a much harder time in the 17th and 18th centuries. Because of the alchemists they had these techniques and a vast array of substances ready and waiting to be used.
These varied from ammonium chloride to various minerals, acids, mercuric chloride, oxides of metals etc.
There is also the harder to estimate expertise associated with manipulating matter, which accumulates with practise and which can be passed on in the way of better furnace designs, the types of crucibles to be used etc. Of course the latter is also where artisans work and alchemists work overlaps. For instance, alchemists and goldsmiths were interested in proportions of matter, e.g. of how much silver or lead was in this gold coin. They were both interested in weighing and the earliest illustration of a balance in a glass case to allow accurate measurement is in a late 15th century alchemical manuscript.
Furnaces and heat were important too, in a more detailed way than to the normal artisan. Alchemists probably helped invent all sorts of useful and not so useful furnaces, and their activities provided the seed or source of various more modern methods of heating used by the early scientists.
The second category was ideas about how the world worked. That is what makes it alchemy, the tying together of practical work with a philosophy of matter, of how and why things worked, rather than the artisanal “I do this, it melts and changes colour and I cast it into the mould, I don’t need to know how and why it does it”.
Which is a bit unfair to artisans, some of whom were alchemists and many of whom must have had their own ideas about how things worked, but because they didn’t write them down or communicate with others and discuss things like that, we don’t really know what they thought.
The alchemists inherited the standard ideas of how the world works of their period, but they discussed them a lot and their variation and how it explained what they observed. For instance the author of the Summa Perfectionis, a 13th century alchemical book, explained the meltings and reactions he saw by means of an early form of corpuscular theory, and other alchemists elaborated upon the four elements theory. A corpuscular theory is, roughly, where everything was made up of little bits of stuff which interacted with each other, and it was important in proto-scientific discussions in the 16/17th centuries.
The alchemists argued about what things were, how they worked etc, and following on from medieval natural philosophy, looked to experiments to demonstrate which view was correct.
Again, without alchemists, the chymists would have had to invent it all themselves, thus retarding the growth of science by a century or two.
Chymist is a word used in the 16/17th century that has been adopted by modern historians of science for the in-between stage where you had real chemistry by modern standards being done, but often based on alchemical works from the medieval period. See for instance Robert Boyle, who was an alchemist, but is also sometimes called the father of chemistry.
As mentioned already, alchemists and physicians were often related to each other. Rhazi, an early Arabic alchemist was a physician, as were many alchemist right up until the 17th century. Medicine and alchemy overlapped at all levels – they were about healing humans or metals, they were about manipulating the four elements to achieve a good outcome, they had similar cosmologies and beliefs in how the world worked. The techniques used were also very similar, or at least physicians were quick to adopt anything alchemists could do, perhaps the most famous example being Paracelsus, who championed mineral and metal based medicines e.g. the use of antimony preparations and I think also sulphuric acid. Moreover it took an education to be properly involved in alchemy, because so much of it was passed on in books, not by word of mouth. The idea that alchemy was an initiatory practise is not wrong, but many, many people practised it without any master telling them what to do, hence the interest in and trade in books of alchemical lore and practises.
So modern chemistry and science in general owes a great deal to alchemy, but it isn’t something that can be pinned down to one specific book, year or person, it was more of a dialogue and foundation digging. Like a child at nursery school – they won’t learn much that is actually correct, and they’ll play with toys, models, things that aren’t of use in the adult world. But they need that development stage in order to go onto bigger and better things.