May a person order food from a vegan restaurant? — assuming you aren't ordering cooked food, to avoid the problem of bishul akum. If your ordering let's say a salad, what could be the problem? There's no problem of bliya (absorption) with the utensils because they only serve vegan food which doesn't contain meat, fish, or milk. And it's raw so it's not bishul akum. Assume it's the diaspora, too.
There are problems with food from a vegan restaurant.
See this Chabad site
A vegan restaurant would not have a hard time getting kosher certification. However, as long as there is no such certification one should not eat there. There are many reasons why a strictly vegan establishment requires kosher certification. Here are a few of them:
It is possible for a minute quantity of animal products to be included in a vegan-certified food. According to the Vegan Society, an outfit which licenses vegan foods, "vegan products must, as far as is possible and practical, be entirely free from animal involvement." Furthermore they state, "Animal products are sometimes used in instances that are not immediately obvious."1
All utensils used to prepare kosher food, as well as countertops, ovens, etc., must be kosher. Meaning, if they were previously used for non-kosher foods, they must be koshered before being used for kosher food preparation.
Wine and grape juice are not kosher unless they are certified kosher (see Wine and Grape Products). Even if the restaurant doesn't have a wine list, many dishes include wine or grape juice in their ingredients.
4. Certain foods must be cooked or baked by a Jew in order to be kosher (seeBaked and Cooked Foods). (Doesn't apply in your case - no cooked food)
And see this site which points out:
Unfortunately for vegans[and Kashrus observers], avoiding all animal byproducts — especially bugs– in the American food system is basically impossible. The Food and Drug Administration even keeps a list of how many bugs and bug parts manufacturers are allowed to have in their products — including many that vegans or vegetarians [and Jews] eat. So how much non-vegan/[non-kosher content] might be in your food? We had a look at the standards and, leaving aside how much animal feces and hair is allowed, compiled 10 of the grossest.
Frozen or canned asparagus Producers are allowed to leave 6 or more attached asparagus beetle eggs and/or sacs on 10% of their spears and either: an average of 40 thrips per 100 grams; or make sure the remaining insects or insect parts have an average aggregate length of 7mm or longer per each 100 grams. Crunchy!
Canned lingonberries They taste so good at Ikea — but that little extra protein comes from the 3 larvae per pound allowed.
Frozen Brussels sprouts Look closely: producers can have up to 30 aphids or thrips in every 100 grams.
Canned orange juice Tangy! Producers are allowed 5 fruit fly and other fly eggs or 1 maggot in each 250 milliliters.
Canned corn Each 24 pounds of corn can have up to two corn ear worms or corn borer larvae less than 3 millimeters or longer, as long as the total larvae, shed skin or fragments thereof does not exceeds 12 millimeters. - See more at: http://www.viciousbabushka.com/2012/03/10-foods-you-thought-were-vegankosher-but-really-arent.html#sthash.YPJJLz1z.dpuf
This article by Nishmat's Rabbi David Sperling explains the issues of non-kosher utensils and bishul akum (2 and 4 from the Chabad site) more in-depth, including when they apply. He also goes into a few other issues such as mar'it ayin, cold and sharp foods, and additional concerns in Israel. (The article isn't specifically about vegan restaurants but is also relevant here.)