Is the water in an aqueous solution undergoing a reaction considered part of the system or surroundings?
The boundary between system and surroundings is defined by you the scientist. Still, some choices make more sense than others, depending on the question you want to answer. When you perform a calorimetric experiment, for instance, you monitor the heat generated or absorbed during a reaction (such as formation of a precipitate). The heat originates or is consumed in the local combination of reagents to form the bonds between precipitate atoms and during segregation of solvent molecules. That heat then dissipates into the rest of the vessel and is registered, for instance as a temperature change. In the final analysis, what you call the system here would probably include only what has changed and you are specifically interested in, namely the molecules involved in the actual reaction. This would include solvent molecules associated with the solubilized reagents (the involvement of solvent is made explicit when reagents are labelled as aqueous, for instance). The behavior of the bulk solvent, which serves an "auxiliary" role, for instance by transferring heat across the vessel, is important during calibrations, but is not the question of interest, and would usefully be regarded as the surroundings.

Water can be regarded as a catalyst in many aqueous reactions, but when a reaction occurs in a solvent that associates strongly "catalyst" is not generally a proper description for the solvent, as the energy of both reagents and products may be stabilized/destabilized by the solvent, and to different degrees (and not just the energy of an intermediate in the reaction).