I am a lifelong vegetarian and a little over 6 months into Starting Strength (now on a Wednesday light day begintermediate program). While everyone can complain about how I should just get over it and eat cows, it's not really that simple to add an entire food group you've never eaten before, and my efforts to change my vegetarianism have resulted in weeks and weeks of food poisoning like symptoms before I decide it isn't really worth it.
Anyway. My personal experience is that it takes a 100% all out effort to get 130g protein per day as a female (which ends up as ~2.5g protein per kg lean body mass, which the 'studies' seem to feel is the minimum end of the optimal range). That's still not the "more ideal" 150g I "should" be eating, but it's the compromise level for me where I don't detect any recovery slowdown, and my gut isn't a mess from too many protein shakes. I forced 150g for 4 weeks just to test it and the only difference in performance I felt was nausea related.
I have to literally track everything I eat every single day and plan all of my meals around high protein foods with very little "free space" for empty calories like, I dunno, white bread. I have become friends with nut flours because you get more protein for the calories from them.
I should note that I don't touch soy or fake meat with a 10 foot pole, so I'm increasing the difficulty level on myself.
I do drink 2 protein shakes a day that are a whey/casein mix. I pair them with a digestive enzyme. The rest of my diet is lots of eggs, greek yogurt, cheese, nuts and seeds, etc. I also notice a huge benefit from supplementing creatine.
So, in my experience, it's possible but an enormous pain in the butt and I wouldn't exactly wish it on anyone. I just think it's unfair to say that it can't be done, especially when vegetarianism is a religion for so many people and if you force them to choose between religion and strength training, guess what they'll choose...