  We study the stability of the Wilson-Fisher fixed point of the quantum
$\mathrm{O}(2N)$ vector model to quenched disorder in the large-$N$ limit.
While a random mass is strongly relevant at the Gaussian fixed point, its
effect is screened by the strong interactions of the Wilson-Fisher fixed point.
This enables a perturbative renormalization group study of the interplay of
disorder and interactions about this fixed point. We show that, in contrast to
the spiralling flows obtained in earlier double-$\epsilon$ expansions, the
theory flows directly to a quantum critical point characterized by finite
disorder and interactions. The critical exponents we obtain for this transition
are in remarkable agreement with numerical studies of the superfluid-Mott glass
transition. We additionally discuss the stability of this fixed point to scalar
and vector potential disorder and use proposed boson-fermion dualities to make
conjectures regarding the effects of weak disorder on dual Abelian Higgs and
Chern-Simons-Dirac fermion theories when $N=1$.
