I've found nuts are basically a super food. No need for refrigeration or preservatives/processing to keep edible. I always have bags of shelled organic nuts on hand and eat handfuls on a daily basis. The only down side I've found is cost; nuts, especially organic ones, are more costly than filet mignon per lb at whole foods.
Note that the first non-meat source (other than fortified breakfast cereal) is baked beans and you'd have to eat at least a full can a day to get your RDI. Or I guess you could eat 2.2kg of yoghurt.
Walnuts have 3mg/100g (33% DV) [1]
Sunflower seeds 5mg/100g (53% DV) [2]
I'm currently holding a 340g bag of raw organic walnuts I opened this morning, and it's nearly empty just from snacking throughout the day. Also snacked on sunflower seeds, but not much.
Dinner was a large bowl of lentils, which have 3.5mg/100g (35%) [3]
Without even optimizing for it I'm over the RDI for Zinc without animal products, this is just what I happen to be stocked up on currently. So clearly it's not much of a challenge for a vegetarian to get sufficient Zinc.
[1] https://en.wikipedia.org/wiki/Walnut#Nutritional_value