(Joint work with Chris Woodward) Consider a Lagrangian submanifold $\bar L$ in a GIT quotient $\bar X = X//G$. Besides the usual Fukaya $A_\infty$ algebra $Fuk(\bar L)$ defined by counting holomorphic disks, another version, called the quasimap Fukaya algebra $Fuk^K(L)$, is defined by counting holomorphic disks in $X$ modulo group action. Motivated from the closed string quantum Kirwan map studied by Ziltener and Woodward, as well as the work of Fukaya--Oh--Ohta--Ono, Chan--Lau--Leung--Tseng, we construct an open string version of the quantum Kirwan map. This is an $A_\infty$ morphism from $Fuk^K(L)$ to a bulk deformation of $Fuk(\bar L)$. The deformation term is defined by counting affine vortices (point-like instantons) in the gauged sigma model, while the $A_\infty$ morphism is defined by counting point-like instantons with Lagrangian boundary condition.