Cool. Well, you'd have to define reproduction. The basic question is: how do molecules make more of themselves? And the answer is: via autocatalysis.
(The following is basic chemistry on rates of reaction, so forgive me if I sound patronizing. I don't intend to.)
Chemicals react at a given rate that varies with certain reaction conditions. Anybody who's mixed vinegar with soda knows that fizzing happens immediately (or to kill it with chemical-speak, large amounts of carbon dioxide are quickly evolved), while on the other hand iron takes days to rust even when it is left out in the open with plenty of water. There are fast reactions and there are slow ones. So how do you speed a reaction up? Essentially, reactions happen more often when the reactant bits (atoms, molecules, or ions) slam each other in the right configuration. So you might simply try to make those bits slam each other more often, which you can do by simply increasing pressure, or temperature, or the concentration of the reactants.
But there's another way: it involves adding a catalyst to the mix. A catalyst doesn't make your reactants collide more often. What happens is when your reactant bits slam into the catalyst, the catalyst essentially rearranges them so that they are in the right configuration, and so while the amount of collisions doesn't change, the proportion of successful collisions does.
So what would happen if you could find a catalyst that could put
itself together? That would be autocatalysis. And we have indeed seen autocatalytic molecules in nature. For example, amino adenosine triacid ester (AATE) molecules, when put in a solution containing raw materials that can be used to construct itself, readily puts them together like any dutiful catalyst would - except what it's making is more of itself, which then catalyses the production of even more AATE, which then ... you get the idea?
AATE is more of a proof of concept; since it's not being used today in any known lifeform, it's highly unlikely that this was actually how life started reproducing. However, RNA also has autocatalytic properties. RNA is known to form "ribozymes", or ribonucleic-acid-enzymes, and some ribozymes also have the capacity to put themselves together. So at least one origin-of-life hypothesis says that the world started out with just RNA - an "RNA world" - and then life developed when the autocatalytic loop widened. Instead of simply chemical A catalyzing its own formation, you get additional steps being put in - chemical A starts to catalyze the formation of chemical B, which catalyzes the formation of chemical C, which catalyzes the formation of chemical D, which then loops right back round and catalyzes the formation of chemical A.
I hope that satisfies your curiosity.