The background is, I am going to write a custom software application for home use, kind of a random picker for good menus for a month.
For that, I am looking for the best dataset you can get for free.
The USDA Food Composition Database is a comprehensive and well-formatted database of nutritional content for various foods. I believe the most recent release was SR28 (Standard Reference 28).
As of 2019, it looks like USDA is setting up a new website for access to this data. You can access data through their API, or you can download the entire dataset for local processing.
You probably want the SR Legacy
or Foundation
data sets. There is no built-in filter for vegetarian or vegan foods so you'll have to filter that yourself, but the vast majority of foundation foods are of plant origin.
If you do the work of classifying foods as vegetarian/vegan, it might be nice to share your results back out to the community through a site like GitHub. I know there are people who would like to build on that work.
The European Food Safety Authority has a similar database, but I was unable to access it at the time of writing this answer.