If I have to pick a restaurant "cold" (without knowing much about it)
I pick Italian or pizza places. Usually you can get a veggie pizza
with no cheese or a pasta dish where you can ask them to leave off
chicken/shrimp, cook with little oil or with wine & garlic, or just
get the plainest marinara sauce on the side.
Japanese food is very iffy if you are vegetarian, be aware that
many sauces are VLF but have fish broth bases. If that is acceptable,
soba or udon noodles are a good lunch choice. Some Chinese places have
steamed dumplings or noodle dishes too, that are ok. Stirfried dishes
are usually very oily and most Chinese cooks will insist you can't cook
it otherwise, even if you request low oil it's not VLF. A Chinese place
near me does a steamed veggie dish but it's extremely plain.