I can't find the text of the original 1992 Midwifery Digest article, however, there is more recent work addressing the same question. It is scientifically dishonest for somebody to cherry-pick that 1992 study and not mention the research that has been done since then.
(Klebanoff et al. 1993) noted that "two recent studies have found that infants who received intramuscular vitamin K were at twice the expected risk for cancer during childhood." They referenced (Golding et al. 1990) and (Golding et al. 1992). All the work since then refers back to Golding et al. as having suggested the association between vitamin K and leukemia. My guess is that it is Golding et al. that was mentioned in the Midwifery Digest as they estimated the incidence between 1.87 and 2.04 per 1000 (matching approximately the 1 per 500 in the claim). (Klebanoff et al. 1993) performed an independent study to examine this hypothesis.
They found "no increased risk of either cancer in general or leukemia in particular among children who received vitamin K during the perinatal period; this finding is in accordance with the observation that the incidence of childhood leukemia in the United States did not increase from 1948 (when few or no newborns received vitamin K) to 1980 (when nearly every newborn received the drug)."
(Vitamin K Ad Hoc Task Force 2003) also addressed the evidence behind this hypothesis. They summarized:
The Vitamin K Ad Hoc Task Force of the American Academy of Pediatrics reviewed the reports of Golding et al and other information regarding the US experience and concluded that there was no association between the intramuscular administration of vitamin K and childhood leukemia or other cancers.
Additional studies that have since been conducted by other investigators have not supported a clinical relationship between newborn parenteral administration of vitamin K and childhood cancer.
They mention specifically (Ross and Davis 2000) that "found no randomized or quasi-randomized evidence of an association between parenteral vitamin K prophylaxis and cancer in childhood. Ten case-control studies were identified, of which 7 found no relationship and 3 found only a weak relationship of neonatal administration of intramuscular or intravenous vitamin K with the risk of solid childhood tumors or leukemia."
The Vitamin K Ad Hoc Task Force also notes that "recent research on the pathogenesis of childhood leukemia additionally weakens the plausibility of a causal relationship between parenteral administration of vitamin K and cancer."
Last, the Cochrane Review of (Puckett and Offringa 2000) stated that "taken as a whole, these studies do not establish a causal relationship between these routes of administration and an increased risk of cancer", and that "no randomized or quasi randomized evidence with regard to the association between parenteral vitamin K prophylaxis and cancer in childhood was found."
Why might the original studies have found an association?
Maybe there was a correlation in the Golding et al. sample that doesn't exist in the full population. There is always a small risk of a false positive. However, a few things stand out to me that might have increased the risk of a false positive in those initial studies.
The original studies (and even many of the subsequent studies) were case-control studies. One draw-back of this type of study "relates to the difficulty of obtaining reliable information about an individual’s exposure status". This was evident in the studies of Golding et al. They admit that "one of the problems encountered in abstracting data from the notes was the fact that the route of vitamin K administration was often not recorded, although vitamin K was noted as having been given." They employed "several strategies [...] to identify the probable route of administration". These were not blinded.
Contrast that with the study of Klebanoff et al., which, while also a (nested) case-control study, used data from a previous prospective cohort study, that had exact records of a child's vitamin K administration (route, dose, name, for example). The later cancer diagnoses were blinded as to the vitamin K exposure status.
Additionally, Golding et al. tested for multiple correlations (at least two: pethidine, and Vitamin K), and reported significance without correcting for multiple hypothesis tests. Test for enough correlations, and you'll find one by chance.
Golding, J., M. Paterson, and L. J. Kinlen. "Factors associated with childhood cancer in a national cohort study." British journal of cancer 62, no. 2 (1990): 304.
Golding, Jean, Rosemary Greenwood, Karen Birmingham, and Martin Mott. "Childhood cancer, intramuscular vitamin K, and pethidine given during labour." BMJ: British Medical Journal 305, no. 6849 (1992): 341.
Klebanoff, Mark A., Jennifer S. Read, James L. Mills, and Patricia H. Shiono. "The risk of childhood cancer after neonatal exposure to vitamin K." New England Journal of Medicine 329, no. 13 (1993): 905-908.
Puckett, R. M., and M. Offringa. "Prophylactic vitamin K for vitamin K deficiency bleeding in neonates." Cochrane Database Syst Rev 4 (2000).
Ross, Julie A., and Stella M. Davies. "Vitamin K prophylaxis and childhood cancer." Medical and pediatric oncology 34, no. 6 (2000): 434-437.
Vitamin K Ad Hoc Task Force. "Controversies concerning vitamin K and the newborn." Pediatrics 112, no. 1 (2003): 191-192.