I see several recipes that call for 'demi-glace' -- so finally I went looking to see if there is a vegetarian or vegan kind (or recipe) out there. (The kinds I normally see are made with things I won't eat--so my search is still on for a kind I might enjoy, instead--even if I have to make it myself.)
I hope I won't get tomatoes thrown at me, but I find the "fancier" French-style cooking too precise and too much work. I much prefer their peasant-style meals that are more rustic and less "pretty" and perfectly cut with too many ingredients. Having said that, when I am looking for a flavor boost, I use Better Than Bouillon stuff in the jars. I can make a more concentrated "stock" or I can use it as salt that adds a bit more flavor. I am not happy that it contains sugar, but for the amounts used, not as bad as some things. They also make mushroom (but I do not know all of the ingredients in there) so may be suitable for vegetarian. If you are looking for the "glaze" and slick look to your food...a quick cornstarch slurry can thicken and gloss foods.
To answer your question, I do not buy nor use demi-glace. I think Knorr makes something but not sure if it is vegetarian (I haven't checked out your link yet).
Thoreau said, 'A man is rich in proportion to the things he can leave alone.'