Constructing an invertible integer matrix given one column.

Yes; the Smith Normal Form algorithm is a sledgehammer that will do this.

More generally, one can transform a column $\pmatrix{a_1\\\vdots\\a_n}$ with integer entries and $\gcd(a_1,\ldots,a_n)=1$ to the unit vector $\pmatrix{1\\0\\\vdots\\0}$ via row operations of the form $R_i\mapsto R_i+mR_j$ ($i\ne j$, $n\in\Bbb Z$), $R_i\leftrightarrow R_j$ and $R_i\mapsto R_j$, each of which corresponds to premultiplying by an elementary matrix with integer entries and determinant $\pm1$. Thus there is a matrix $M$ integer entries and determinant $\pm1$ with $M\pmatrix{a_1\\\vdots\\a_n}=\pmatrix{1\\0\\\vdots\\0}$. Then $M^{-1}$ also has integer entries, and determinant $\pm1$ and its first column is $\pmatrix{a_1\\\vdots\\a_n}$.