For a binary integer program (IP) $\max c^\T x, Ax \leq b, x \in \{0,1\}^n$, where $A \in \R^{m \times n}$ and $c \in \R^n$ have independent Gaussian entries and the right-hand side $b \in \R^m$ satisfies that its negative coordinates have $\ell_2$ norm at most $n/10$, we prove that the gap between the value of the linear programming relaxation and the IP is upper bounded by $\operatorname{poly}(m)(\log n)^2 / n$ with probability at least $1-2/n^7-2^{-\operatorname{poly}(m)}$. Our results give a Gaussian analogue of the classical integrality gap result of Dyer and Frieze (Math. of O.R., 1989) in the case of random packing IPs. In constrast to the packing case, our integrality gap depends only polynomially on $m$ instead of exponentially. Building upon recent breakthrough work of Dey, Dubey and Molinaro (SODA, 2021), we show that the integrality gap implies that branch-and-bound requires $n^{\operatorname{poly}(m)}$ time on random Gaussian IPs with good probability, which is polynomial when the number of constraints $m$ is fixed. We derive this result via a novel meta-theorem, which relates the size of branch-and-bound trees and the integrality gap for random \emph{logconcave} IPs.