Well, In-N-Out does have a sort of cult-like following, that's part of the backlash, right? And the fact that they take pride in their very tiny menu gives them a stronger practical defense, I think, than someone like McDonald's who will clearly (and wisely) find menu space for whatever moves product.
As for "why would a vegan want to go to a place like In-N-Out Burger" -- I think there are two answers. First is the specific -- I personally LOVED In-N-Out prior to my dietary change (more than 5 years ago) and I sincerely would be thrilled if I could get the attention to detail they offer their products to something I could/would eat now. If they made a vegan-friendly burger with fresh toppings that was parallel to their outstanding fast food burger -- I'd greedily become a huge fan and customer when I could.
The second is the broader argument. Plant-based dieters are a growing share of the overall market, and it's not unreasonable to ask for growing respect on the menu. If I get a Subway sandwich, I end up basically ordering a "black forest ham and cheese, hold the ham and cheese" but paying full price, since there's no discount (there are several lowest-price subs there, the closest they make to an accommodation is putting double cheese at no charge on your "veggie delite" sandwich).
So... no, I am not mounting a civil right campaign decrying unfair treatment. But I do _want_ Subway to offer falafel (and why the fuck not, by the way) or a vegan friendly protein of some sort. And when i call places in advance to ask about the menu, I usually let them know if I won't be coming there due to my misgivings (lard in the beans, chicken in the rice, I'll get my Mexican food elsewhere).
Maybe some think that makes me an activist/terrorist, but I don't think so. It's just very small-scale agitation for positive change.