There is a great deal of difference within vegetarianism, and it is worth distinguishing three common positions.
Vegan: no animal products at all, the lowest carbon footprint, and the standard we require on every journey. Lacto-ovo vegetarian: includes eggs and dairy, with a footprint roughly 1.5 times a vegan one, mostly at the agricultural stage. Convenience vegetarian: vegetarian occasionally and meat-based most of the time, with a footprint close to an ordinary diet and essentially no carbon saving.
One dimension that is often overlooked is transport and packaging: a locally grown, in-season lacto-ovo meal may have a lower footprint than vegan ingredients flown in from abroad. So local and vegan matter equally.
TeaVegVoyage insists on 100 per cent vegan, not because lacto-ovo vegetarianism is bad, but because insisting is itself a choice: if you are going to do it, do it at the level that needs no explanation.
One practical line to take away: rather than agonising over labels, remember local, in season, little packaging. Their effect on your footprint is often larger than the vegetarian category itself.