We consider the problem of jointly recovering the vector $\boldsymbol{b}$ and the matrix $\boldsymbol{C}$ from noisy measurements $\boldsymbol{Y} = \boldsymbol{A}(\boldsymbol{b})\boldsymbol{C} + \boldsymbol{W}$, where $\boldsymbol{A}(\cdot)$ is a known affine linear function of $\boldsymbol{b}$ (i.e., $\boldsymbol{A}(\boldsymbol{b})=\boldsymbol{A}_0+\sum_{i=1}^Q b_i \boldsymbol{A}_i$ with known matrices $\boldsymbol{A}_i$). This problem has applications in matrix completion, robust PCA, dictionary learning, self-calibration, blind deconvolution, joint-channel/symbol estimation, compressive sensing with matrix uncertainty, and many other tasks. To solve this bilinear recovery problem, we propose the Bilinear Adaptive Vector Approximate Message Passing (BAd-VAMP) algorithm. We demonstrate numerically that the proposed approach is competitive with other state-of-the-art approaches to bilinear recovery, including lifted VAMP and Bilinear GAMP.