What tests do I need to perform for VECM and VAR to be considered robust? I know LM test for residual autocorrelation is mandatory, but what about Jarque-Bera test? Is that necessary?And what should I do if my regressions don't pass that test?
Jarque-Bera is not mandatory for either VAR or VECM. Not passing a test for normality (or, more precisely a test for symmetry and no excess kurtosis) at least asymptotically has no implications on the validity of either tests or estimators in VECMs.
It is true that, e.g., Johansen derives MLEs under the assumption of a normal likelihood (i.e. normal errors), but then derives the large-sample distribution of his tests/estimators under much broader moment conditions. In the Palgrave handbook of econometrics, Johansen writes:
Thus the limit results hold for i.i.d. errors with finite variance, and not just for Gaussian errors.