I was imagining a scenario where an 21st century engineer accidentally finds himself transported through time to early Renaissance Europe. Realizing that no help or rescue is on the way, he decides to use his knowledge to become an influential businessman. It occurred to me that rubber has many modern uses. Is there any way a usable source could be obtained prior to the discovery of the Americas?
There are some 200 species of plants that produce latex.
The major and common ones are:
Euphorb family (Euphorbiaceae)
milkweed family (Asclepiadaceae),
mulberry family (Moraceae),
dogbane family (Apocynaceae), and
chicory tribe (Lactuceae) of the sunflower family (Asteraceae).
The problem is that many of these plants are not native to Europe or would not be found in any real quantity.
Most can be found in hot places. The closest you will find would be dandelion (Lactuceae). Which while common in Europe, the amount you would need in order to make rubber would likely be staggering.
But dandelions are your best bet! Here's a handy link on how to use it and the process.
There was an attempt in WWII to use milkweed as a source of rubber because of War time shortages but the percentage (1-2%) was simply too small.
Today, we haven't succeeded at making rubber from dandelion and are working to breed some with more latex in them. The Medieval breeds most likely were even lower in percentage than the specialized breeds of today.
However, if you look at the link on how to make latex rubber at home, you'll see that while it is not commercially viable and can't be used in exactly the same way as today's rubber, it can be used a number of different ways in applications such as water proofing. It's not good enough to make, say a tire, but it's something.
Taraxacum kok-saghyz "is a species of dandelion native to Kazakhstan that is notable for its production of high quality rubber". It can be grown very easily almost everywhere in Europe. The Soviets have actually performed large-scale experiments, before and after the war, achieving "200 kg of rubber per hectare". The efforts to grow kok-saghyz did not continue after the war because in peacetime ordinary natural rubber is much cheaper. That's what Wikipedia says.
There are two major sources of rubber that you could have access to. But you would have to bankroll or lead an expedition to (tropical) West Africa or the Congo. In west Africa you've got Lagos Silk Rubber (Futumia elastica), and in the Congo you've got Landolphia Rubber (Landolphia genus).
When it was discovered Lagos Silk Rubber was over exploited and almost driven extinct but that won't be until 1894 so it will still be a viable source during the Renaissance. Landolphia Rubber and the activities of Belgium to get it is a rather horrific tale but again the Belgian colony won't be established until 1865. Either way it will require hard work to get.
As a bonus there is also the potential to get gutta-percha via the silk road. The chemical name for rubber is cis-1,4-polyisoprene, gutta-percha is the trans variant of this molecule. It's more rigid than rubber but it is still a thermoplastic in time when such a product is not commonly available. It comes from the Palaquium genus and is found in Malaysia. You would just need to get someone in China to acquire it, though that may be easier said than done.