What makes humans omnivores, and not herbivores?
As the answers given already point out, humans are generally considered omnivores that are able to use food sources available to them opportunistically. Apart from that, there actually is not much to debunk in the PETA statement - it is just a slightly one sided view on the history of human diet. Just as one sided as the claims that humans would not have evolved a really big brain without consuming animal products. Especially the canine teeth cannot be seen as evidence for prehistoric hunting, as in great apes the size of canine teeth is ridiculously well explained not only by defensive but mostly by mating behaviour; apart from that, humans do not even have those, and that most likely is caused by the trend towards monogamy which reduces inter-male aggression. 

However, coming back to the actual question, in the relative short history (approx. 200-300ky) of our species there is evidence for 'several transitions towards or away from a vegetarian diet in humans' (Nielsen et al., 2017).

And in fact, there even is a human-specific evolutionary derived haplotype (younger than the Human-Neandertal split) that shows signatures of positive selection and is associated with increased efficiency in the synthesis of long chain fatty acid from plant precursor molecules - which would not really be necessary for meat eaters, as these fatty acids are readily available in animal products (Ameur et al., 2012). More specifically, the genes lying on that haplotype belong to the FADS cluster that seemed to have had adaptive value - also for synthesising long chain fatty acids from medium chain plant precursor fatty acids - in African populations even more recently (approx. 85 kya, Mathias et al. 2012).

This finding has been supported by a more recent study by Kothapalli et al. (2016). They investigated a 22bp-indel allele in the FADS2 gene that is associated with both higher blood levels of long chain fatty acids and exhibits high frequency (signature of positive selection) especially in South Asia and Africa where there is long tradition for plant-based diet.



Conclusion:

Modern humans (and this is relevant, as diet varies strongly between species and therefore the species comparison does not need to be very informative) appear to have changed diet frequently but there is evidence for human-specific molecular adaptations towards plant-based diets.