Why are the Trappist monks vegetarians?
The short answer is that it started out as a monastic custom.
Simply put, it is because the originally in the Rule of St. Benedict (516), monks were not to eat meat. St. Benedict started a monastic tradition in this regard. He, nevertheless, allowed sick monks to eat the flesh of animal until their health was restored. It is explained in Chapter 39 of the Rule of St. Benedict.
Chapter XXXIX: Concerning the Quanity of Foods
We believe that for the daily refection in all the months of the year, alike when it is at the sixth hour of the day as when it is at the ninth, two cooked dishes will avail, in consideration of the weakness of different individuals, that he who perchance cannot eat of one may be sufficiently catered for by the other; so for all the brethren let two cooked dishes suffice; and if there be fruit in addition or young vegetables let there be added a third dish also. Of bread let one pound by weight suffice, whether there be but one meal, or both dinner and supper, though if they are going to sup let a third part from that same pound be kept back by the cellarer and served when they sup. But if by chance any hard work shall have been done, it shall be within the discretion and power of the abbot to make some addition, if it be expedient, so long as all surfeiting be avoided and he take care that indigestion never overcome the monks; for nothing is so adverse to any Christian as surfeiting, as says our Lord: “See to it that your hearts be not weighed down with surfeiting.” And let not the same quantity be served to boys of tender age, but less than to their elders, moderation being observed in all cases. And let all abstain entirely from the eating of the flesh of quadrupeds, altogether excepting from this rule the weak and the sick.
If memory serves me correctly, even the sick in Trappist monastic communities do not eat meat.
Trappists are not the only Catholic Religious that do not eat meat. Some traditional Poor Claires are also vegetarians, as was their original custom. I am confident that you will find other Catholic Religious who are vegetarians!
The following may be of interest: