The sources I find suggest that the validity of the theory is still up for debate (surprisingly).
This NY Times article dismisses the theory:
Later, other photographs were taken by W. H. Sheldon, a researcher who believed that there was a relationship between body shape and intelligence and other traits.
Mr. Sheldon has since died, and his work has long been dismissed by most scientists as quackery.
and there's a more detailed, academic critique here.
At the same time, academic researchers are conducting empirical studies as recently as 2013 that appear to confirm Sheldon's hypothesis:
The questionnaire was administered to 242 (two
hundred and forty two) University of The Gambia students by stratified sampling technique based on body types
(whether endomorph, mesomorph or ectomorph). The physical body type an individual possesses has an
influence on the behaviour or personality of that individual, as posited by Sheldon’s constitutional theory. This
was confirmed, even when applied in the African society specifically in The Gambia, by this study. [emphasis mine]
Here's a 2006 paper making similar claims:
Investigated whether measures of personality considered compositely and individually differ significantly among groups differentiated according to self-perceived somatotype (PSS). The Bodv Cathexis Scale, Eysenck Personality Inventory, and Tennessee Self-Concept Scale were employed to assess personality in 285 college males, and the indices of somatotype were measured using the Perceived Somatotype Scale. MANOVA revealed significant differences in global personality among the groups. Univariate analyses indicated that the PSS groups differed significantly in self-concept, extraversion, neuroticism, and body cathexis, while the PSI groups differed significantly in extraversion. Apparently, the male personality is partly a function of the body build perceived as self, the image viewed as ideal, and whether a discrepancy exists between the figures perceived as self and ideal.
And here's a third, from 2010:
After examining the relativity between personal measurement categories of the subjects and character type through the MBTI, Extraversion(E) type showed inverse correlation in stature, height, biacromial breadth, and subcutaneous fat thickness of posterior iliospinale. Contrary to the E type, the Introversion(I) type showed positive correlation. Sensing(S) type showed inverse correlation only in biacromial breadth, Intuition(N) type showed positive correlation in most categories including height, width and the subcutaneous fat thickness. Thinking(T) type showed positive correlation in bust point-bust point, chest depth and hip width. Feeling(F) type, on the other hand, showed inverse correlation. The Judging(J) type showed inverse correlation in stature, height, length and the parts of chest. In contrast, Perceiving(P) type showed positive correlation in other categories including biacromial breadth, same as the J type.
A Google scholar search for "Somatotype personality", limited to publications since 2010, produces almost 500 results, which suggests this is still an active area of research.
Additional caveat: I am not qualified to assess whether the research present is correcting for discipline appropriate factors, and thus whether their findings are valid. However, taken at face value, it looks like there is some relationship between somatotype and personality, and all of these papers are referencing the theory. Some of them are using more detailed measurements however.
EDIT: Some additional information in response to one of the comments: Yes, one's somatotype can change over time, for at least some groups (I speculate: for anyone). For example, see this study, in which 67% of subjects changed dominant somatotypes during puberty.