Is it possible to have Zero order Elementary Reaction?
While zero order reactions are certainly possible and have considerable technological use, for example, in breathalysers used to determine alcohol content in ones breath, the question is what do you consider an 'elementary' reaction to be ?

In a pure sense the only 'elementary' reactions that occur are unimolecular and bimolecular ones (possibly also termolecular but exceptionally rare). A unimolecular reaction could be a bond dissociation or cis-trans isomerisation which can occur in an isolated molecule. Bimolecular reactions such as atom/molecule &amp; molecule/molecule reactions are very common and strictly involve only two species. 

So in this sense zero-order reactions have to have a more complex nature, such as involving an enzyme as in alcohol decomposition.

(Note also that the order of a reaction is measured experimentally as the sum of the powers to which reactants are raised in the rate expression, $rate= kA^aB^bC^c$ the order is $a+b+c$. Often it is the case that an apparently simple reaction such as $\ce{H2 + I2 }$ is only fully understood by a complex series of reactions, in this case the rate expression has fractional and reciprocal orders. The molecularity m refers to the coefficients in an elementary reaction, for example $m = a + b$ if the reaction actually occurs in a step $aA+bB \rightarrow$ so that m is a small positive number usually $1$ or $2$, for example $ m=1;\, A\rightarrow product$ or $ m=2;\, A+B\rightarrow product$)