Density functional theory (DFT) calculations confirm that the structurally ordered ${\mathrm{Fe}}_{2}\mathrm{VAl}$ Heusler alloy is nonmagnetic narrow-gap semiconductor. This compound easily crystallizes in various disordered phases with high concentration of antisite defects. We study the effect of structural disorder on the electronic structure, magnetic, and electronic transport properties of the full Heusler alloy ${\mathrm{Fe}}_{2}\mathrm{VAl}$ and its off-stoichiometric equivalent ${\mathrm{Fe}}_{2}{\mathrm{VAl}}_{1.35}$. Data analysis in relation to ab initio calculations indicates an appearance of antisite disorder mainly due to Fe-V and Fe-Al stoichiometric variations. The data for weakly magnetic ${\mathrm{Fe}}_{2}{\mathrm{VAl}}_{1.35}$ are discussed with respect to ${\mathrm{Ni}}_{2}\mathrm{VAl}$. ${\mathrm{Fe}}_{2}{\mathrm{VAl}}_{1.35}$ can be classified as a nearly ferromagnetic metal with a pronounced spin-glassy contribution, which, however, does not have a predominant effect on its thermoelectric properties. The figure of merit $ZT$ is at 300 K about 0.05 for the Fe sample and 0.02 for Ni one, respectively. However, it is documented that the narrow $d$ band resulting from Fe/V site exchange can be responsible for the unusual temperature dependencies of the physical properties of the ${\mathrm{Fe}}_{2}{\mathrm{TiAl}}_{1.35}$ alloy, characteristic of strongly correlated electron systems. As an example, the magnetic susceptibility of ${\mathrm{Fe}}_{2}{\mathrm{VAl}}_{1.35}$ exhibits singularity characteristic of a Griffiths phase, appearing as an inhomogeneous electronic state below ${T}_{G}\ensuremath{\sim}200$ K. We also performed numerical analysis which supports the Griffiths phase scenario.