The problem of finding a path between two points while avoiding obstacles is critical in robotic path planning. We focus on the feasibility problem: determining whether such a path exists. We model the robot as a query-specific rectangular object capable of moving parallel to its sides. The obstacles are axis-aligned, rectangular, and may overlap. Most previous works only consider nondisjoint rectangular objects and point-sized or statically sized robots. Our approach introduces a novel technique leveraging generalized Gabriel graphs and constructs a data structure to facilitate online queries regarding path feasibility with varying robot sizes in sublinear time. To efficiently handle feasibility queries, we propose an online algorithm utilizing sweep line to construct a generalized Gabriel graph under the $L_\infty$ norm, capturing key gap constraints between obstacles. We utilize a persistent disjoint-set union data structure to efficiently determine feasibility queries in $\mathcal{O}(\log n)$ time and $\mathcal{O}(n)$ total space.