We prove new bounds on the additive gap between the value of a random integer program $\max c^Tx,\ Ax\leq b,\ x\in\{0,1\}^n$ with $m$ constraints and that of its linear programming relaxation for a wide range of distributions on $(A,b,c)$ . We are motivated by the work of Dey, Dubey, and Molinaro (SODA '21), who gave a framework for relating the size of Branch-and-Bound (B&B) trees to additive integrality gaps. Dyer and Frieze (MOR '89) and Borst et al. (Mathematical Programming '22), respectively, showed that for certain random packing and Gaussian IPs, where the entries of $A,c$ are independently distributed according to either the uniform distribution on $[0,1]$ or the Gaussian distribution $\mathcal{N}(0,1)$, the integrality gap is bounded by $O_m(\log^2 n / n)$ with probability at least $1-1/n-e^{-Ω_m(1)}$. In this paper, we generalize these results to the case where the entries of $A$ are uniformly distributed on an integer interval (e.g., entries in $\{-1,0,1\}$), and where the columns of $A$ are distributed according to an isotropic logconcave distribution. Second, we substantially improve the success probability to $1-1/poly(n)$, compared to constant probability in prior works (depending on $m$). Leveraging the connection to Branch-and-Bound, our gap results imply that for these IPs B&B trees have size $n^{poly(m)}$ with high probability (i.e., polynomial for fixed $m$), which significantly extends the class of IPs for which B&B is known to be polynomial. Our main technical contribution is a new linear discrepancy theorem for random matrices. Our theorem gives general conditions under which a target vector is equal to or very close to a $\{0,1\}$ combination of the columns of a random matrix $A$ . The proof uses a Fourier analytic approach, building on work of Hoberg and Rothvoss (SODA '19) and Franks and Saks (RSA '20).