Fjällräven is a very well-established Swedish brand that makes exactly this, and even have their own brand of wax. As far as I know, none of their products have any synthetic membranes.
Here in Denmark, "Fjällräven" is a household name to the point that it can be used to refer to their small backpack model.
Their waterproofing wax is a mix of paraffin and bee wax. So if the whole point is to avoid petrochemicals or avoid animal products, this isn't a solution.
A combination of beeswax and microcrystalline wax (made from petrolatum) works the best, but beeswax alone will provide fine water repellency, just slightly less durability and ease of application.
Here in Denmark, "Fjällräven" is a household name to the point that it can be used to refer to their small backpack model.
[0]: https://www.fjallraven.com/