Given a space $\mathcal{U}$ of dimension $\leq n-1$ and set of integer vectors $\mathbf{V}$ which span dimension $>n-1$ inside $\Bbb Z^m$ where $m\geq n$ holds is there always an integer vector $v\in\mathbf{V}$ such that for all integer vectors $u\in\mathcal{U}$ we have $\langle u,v\rangle\neq0$?
What is the probability of above happening if $\mathcal U$ is fixed while $\mathbf V$ is uniformly randomly picked with all coordinates within a bound?
As stated the answer is trivially no: take $U$ to be the span of the first $n-1$ standard basis vectors, and take $V$ to be the set of the first $n$ standard basis vectors. Perhaps you intended for $V$ to be an actual sublattice, not just a set of vectors whose span is large. But the answer is still no: take $U$ as before, and take $V$ to be the span of all the standard basis vectors except the first one (that is, $V$ is the set of all integer vectors whose first coordinate is $0$; this meets your dimension requirements when $m\ge n+1$, but every vector in $V$ is orthogonal to the first standard basis vector.