Do humans produce rennin?
Scanning various reviews it seems that everyone who mentions the possibility of a human chymosin refers to a single paper. So for example this 2014 review has a single reference to a human chymosin:


  Henschel et al. detected a
  protease in the gastric aspirates of newborn infants within 6–10 h postpartum that was not
  pepsin [62]. The electrophoretic mobility and immunoreactivity are similar to that of calf
  chymosin, a protease that cleaves κ-casein and causes casein curdling. This protease is
  unique in that it disappears from gastric fluid at 10 days postpartum and is not found in adult gastric fluid.


I don't have access to the Henschel et al. paper (from 1987) but here is the abstract:


  The electrophoretic mobilities of proteases present in gastric juice taken within 10 h of birth from 5 healthy, premature infants were compared with calf chymosin, pig pepsin A and human adult gastric juice. The juice from 2 infants contained predominantly a chymosin-like enzyme, another had almost exclusively pepsins similar to those of the adult juice, while the other two contained a mixture of both. The pepsins consisted of two elements, probably pepsin A (EC 3.4.23.1), and pepsin C (EC 3.4.23.3). Single radial immunodiffusion gave a definite reaction to calf anti-chymosin serum in five samples taken from a further 17 infants. These results indicate that some human infants secrete chymosin. The reaction in the immunodiffusion assay indicated a much lower enzyme activity than that implied from electrophoretic separations. It is suggested that species differences resulted in poor cross-reactivity of the antiserum.


Now, obviously, without seeing the data it isn't possible to be conclusively critical, but the quality of the evidence seems to be rather weak, being based upon similar electrophoretic mobilities and an immunodiffusion assay (why not a Western blot?) with an overt apology for weak cross-reactivity of the antiserum used.

However, leaving all of that to one side, the strangest thing about this is the sporadic appearance of the proposed chymosin: although 4/5 were scored as chymosin positive in the 1st experiment, is it likely that 2 of these would apparently contain little pepsin? And in the second experiment (the immunoassay) only 5/17 scored positive for chymosin.

Evidently no-one has ever reproduced this result, and the evidence for the pseudogene (but no active gene) is very strong. I vote that humans do not produce a chymosin.

Update
Having read Bryan Krause's answer: if the human gene product was lacking an exon's worth of amino acid sequence then presumably it wouldn't have an electrophoretic mobility that was closely similar to the calf protein.