Beside total proteins and the need for protein combining (see this question), here I ask a more specific question regarding single amino-acids: what are the most likely essential amino acids to be deficient in a vegan diet? Where to find them?
A vegan diet is based only on vegetal foods. If we analyze the amino acid content of different food protein sources (animal and plant proteins), lysine is consistently at a much lower concentration in all major plant-food protein groups than in animal foods (1, 2). Lysine is one of the essential amino acids, thus lysine is most likely to be the first limiting amino acid in diets that are based predominantly on cereal grains (2).
For this reason the American Dietetic Association, in its 2009 position paper on vegetarian diets, recommends an increase of consumption of beans and soy products in order to satisfy the essential amino acids requirements in vegan diets (3).
This quote from a recent study responds your question very clearly:
Amino acid intakes are sufficient and lysine intake might only be limiting in vegan individuals who have a low protein intake when basing their diet on a very limited and monotonous pattern where the protein intake would only come from, for example, grains alone—an unrealistic situation in developed countries.
I would like to point to my answer to a related question, to highlight methionine as a potential deficient amino acids in certain diets. This article discusses how high glycine in vegan diets may lower bio-availability of methionine, but at the same time, lower methionine levels seem beneficial.
I would also point to this interesting result by another study assessing the diet of 36 vegans, mentioning:
The present study also showed that the lower dietary intake of AAs in vegans was hardly reflected in the plasma concentrations of free AAs.
However, the latter study also mentions a tendency for lower intake of lysine by vegans (also of methionine, but to a much lower extent), which may seem to contradict the first statement of this answer.
To recap, I think I showed there is some controversy and the results of the scientific inquiries into vegan diets may vary. Also, important, intake deficiency does not translate into serum level deficiency. A balanced vegan diet (beans, grains, legumes) should not be lacking any essential amino acids. An unbalanced one (e.g. grain-based) could be lacking in lysine (perhaps methionine, in special cases, e.g. no soy intake), so I would bet on lysine for 'the most likely deficient amino acid in vegan diets'. To find it, it takes just to only eat a variety of vegan products.