Are there large integer matrices with entries computable in polynomial time, such that all minors are nonzero?

For rational entries, a Cauchy matrix works, e.g. $a_{ij} = 1/(2^n+i-j)$.

For integer matrices, pick a prime $p > 2^{n+1}$, and let $a_{ij}$ be the smallest positive residue of $1/(2^n+i-j) \bmod p$, which can be computed in polynomial time by the extended Euclidean algorithm. By the formula for a Cauchy determinant, all minors are nonzero modulo $p$, so they remain nonzero as integers.

A $2^n \times 2^n$ Hadamard matrix, see https://en.wikipedia.org/wiki/Hadamard_matrix, has entries $\pm 1$ and is equal to its transposed inverse (up to a scalar factor $2^n$). Thus all its minors are nonzero.

The Wikipedia article contains Sylvester's contruction of a $2^n \times 2^n$ Hadamard matrix. With the information in that article it is easy to see that an entry of such a matrix can be computed in time $O(n)$.