Historically, the vitamin concept arose out of nutritional studies that demonstrated the dietary requirements of additional factors not supplied by a diet of protein, carbohydrate and fats. These substances are 'specific in function' and are required by the body in very small amounts, and were termed vitamins by Casimir Funk (see Hopkins' Nobel Lecture).
One substance that comes very close to being a vitamin for another organism but not for humans is biopterin, although I suspect that the term 'essential growth factor' is more appropriate than 'vitamin', given the current state of biochemical knowledge.
The organism is the protozoan Crithidia fasciculata, which requires the unconjugated pteridine biopterin as growth factor (Patterson et al).
Despite this requirement, and despite the fact that this metabolic need is the basis of an assay for unconjugated pteridines (Dewer and Kidder), no one knows the molecular basis (although there are some suggestions that is it required in pyrimidine biosynthesis) (see Kaufman)
Dewer and Kidder go further and state that "the only organisms at present known to have a dietary requirement for an unconjugated pteridine are members of the flagellate order Kinetoplastida (trypanosomids)" although personally I have never been able to confirm this except for Crithidia. Trypanosomes (the causative agent of sleeping sickness), for example, have never been shown, as far as I am aware, to require an unconjugated pteridine.
Humans, of course, do require an unconjugated pteridine (in the form of tetrahydrobiopterin, or BH4): it is the cofactor for phenylalanine hydroxylase, tyrosine hydroxylase, trytophan hydroxylase, for an enzyme involved in the oxidation of glyceryl ethers, and for nitric oxide synthase (among other roles).
However, it is not a vitamin! We can synthesis it (from GTP, as it so happens). This aspect is discussed in detail by Kaufman, as is the essential role of BH4 in neurotransmitter biosynthesis.
None of the above enzymic systems are known to occur in Crithidia (Kaufman). This organism, for example, cannot convert Phe to Tyr: the only enzymic system, or enzymic systems, capable of converting Phe to Tyr require an unconjugated pteridine (no cytochrome P450 system, for example, is capable of this transformation).
Both plants and E.coli are also of interest.
The evidence for any BH4-requireing enzyme in plants is scant. Plants can synthesise Phe and Tyr (they have the enzymes of the shikimate pathway) but they cannot convert Phe to Tyr. Phenylalanine hydroxylase, for example, has never been isolated from a plant source.
The same applies to E.coli. This organism has no requirement for an unconjugated pteridine and cannot convert Phe to Tyr (see Miller and Simmonds and Kaufman).
But this is not the case for all bacteria. Those, such as Chromobacterium species (ref), that are capable of using Phe as a carbon source have phenylalanine hydroxylase, which requires BH4 (or a closely related pteridine) as cofactor. In these cases, too, the pteridine (BH4?) is not a vitamin but is synthesized de novo from GTP.
Thus, to summarize. The only organism known to require an unconjugated pteridine as a dietary growth factor is Crithidia fasciculata but no one knows why. You and I (as humans) require BH4 as cofactor for key hydroxylation reactions, including the conversion of Phe to Tyr, but it is not a vitamin: we synthesis it. Plants and E.coli have no known requirement for biopterin (or any unconjugated pteridine) and cannot convert Phe to Tyr. Some bacteria, however, do require BH4 (or a closely related pteridine) as cofactor for hydroxylation reactions but these organisms also synthesis it.