Unfortunately, I do not have most of my sources on agriculture on hand right now, so I am quoting from Ukrainian theses/articles on agriculture (e.g. Nesterchuk 2017 or Brazhenko, Brazhenko, Gangur and Chekrizov), which reference the historical theses (specifically, I do not at present have direct access to either Vavilov or Wolf, and also not Bodganovych) and are available online. Please note this when citing further. Also keep in mind that this is imperial Russian/Soviet/post-Soviet history, so any borrowings from Native American agriculture could have been left out or lost.
The historical narrative repeated in Ukrainian theses and articles on agriculture is that the sunflower was imported into Ukraine via Spain (the import of 1510, known as the Peruan Chrisantemum back then). It came to Ukraine in the 18th century, first as a decorative plant. However, according to Vavilov (cited in Nesterchuk 2017), Ukraine and Russia were the first countries to use the sunflower as a major source of vegetable oil, and the sunflower as popular oil culture spread from these countries rather than the other way around. By the mid-19th century it is already very much a feature of local agriculture (e.g. there is a mention (Bogdanovych 1877) that it is a crop that is rather taxing on the soil and is therefore largely cultivated on rented land, and the first oil processing plant was built in the mid-19th century). The early 20th century saw a lot of successful experiments in selection under such scientists as V.S. Pustovoitov in order to improve the sunflower's capacity for oil production.
So, if Vavilov and Nesterchuk are to be believed, there was a general explosive increase in the production of sunflower oil in the 20th century, but as far as oil-bearing varieties/processing techniques went, this popularity originated in Eastern Europe. I will add as an observation that even now, sunflower oil retains a place in Ukrainian and Russian cuisine as the frying oil of choice for most Ukrainian households, taking up 70% of the oil-producing crops in contemporary Ukraine and up to 90% of produced oil (statistics from Nesterchuk 2017 and Kohan et al. 2015). There is a lot of work being done in the agrarian sector in order to improve the productivity/lessen the drain on the land via crop rotation (for an example, see Brazhenko et al. 2006 and Kohan et al. 2015. Note that some of the co-authors from the 2006 paper are also present in the 2015).
Literature: