Claiming you can't be vegan because you are poor makes perfect sense, but I would argue that you can still be vegetarian at least and reduce dairy and cut out eggs. By all means avoid the vegan cheese and almond milk, but being poor doesn't stop you buying fruit and veg and beans and rice and pasta and bread.
Looking at my own life, I first focused on doing things for myself and family such as a good job and salary, a house and car and so on, and the ethical lifestyle to a large extent came later on. I think if I had grown up in a much less privileged background I might never have given up meat at all (or at least not yet), because I might never have set aside the time and money to read various books on vegetarianism.
However, conversely, it's also true that now I have understood the ethics I would not go back to eating meat now even if I was very poor and in a difficult situation in life.
I think Jack's statement "Poor people don’t need to be moralised to about what they are eating" is dubious, though. And it shows in the way she supports this claim with a weak argument about not taxing beans. The use of "moralised to" as a negative is perhaps reminsicent of negative statements like "do-gooder" and "moral high ground". How can these by negative statements? There is a fundamental problem in society that people who try to live more moral lifestyles are implicitly criticised and the true goal is to try and avoid not just a morally bad lifestyle, but also to avoid a morally good one. Striving for mediocrity in ethics is implicitly supposed to be the goal, and fitting in with the majority effectively trumps striving to be the best you can be.
I think, at least in theory, we don't want to dismiss under privileged people as being incapable of assessing moral arguments and vegans should educate all society about veganism.
However, in practice I think you do need to know when to pick your battles. That paragraph I wrote above may be a reasonable discussion point for this forum, but I am not going to react like that to an actual poor person telling me they can't be vegan. I would probably just sympathise and maybe just suggest that they try to reduce factory farmed meat if possible, or maybe not even that, maybe even agree with them or change the subject. Depends on the situation or even how I felt. So I think Jack's article has an important truth to it as well, at least in practice. Even though strictly speaking I tend to disagree with the thrust of it, ranting angrily in opposition to it is just going to make vegans look bad again.