For example Ivan Kolar worked on this issue but on functeurs with general fibers.
Have you checked Chapter V in Kolar-Michor-Slovak: Natural Operators in Differential Geometry? I think they generalize 'classical' results (like Epstein's and Thurston's, that locally determined functors are in fact determined by jets of a certain order), but, despite their fairly general assumptions, also treat special cases of spaces or fibres -- and they also give a short summary of related results at the end of the chapter. (I don't know whether this is helpful; I just happened to have a copy of the book.)
Thanks for your answer. Efectively, I have got this book but I am finding more recent results if that exists. I am specially interesed in the vector bundle functors for extend some results over the tangent bundle functor.
As a first iteration, I would try this http://dml.cz/bitstream/handle/10338.dmlcz/107982/ArchMathRetro_042-2006-1_7.pdf ... It took me to this one http://journals.impan.pl/cgi-bin/doi?ap82-3-6 where is a reference [6]. This might be what you are looking for. However, the fiber preserving only. When more general, this should be mentioned in [6] (Kolar, Mikulski DGA 1999. I guess the biggest group working in this area contains Doupovec, Kurka, Miukulski and Kolar (they should be active in this field...as far as I know). Of course, it depends in waht extent functors you want to classify ... and how fine and explicit is the classification-- often thse are done using the algebraic (Weil algebra) or vector space data.
Thanks Svatopluk Krýsl. I was looking the works of that group. I found a lot of very interesting information. I am doing the classification by Weil algebras but I am interested in a finer one.
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