Yes, even if water is at thermal equilibrium with the surrounding air and not exposed to any external heat source, we still say that water gains heat from the surroundings to evaporate. This is because evaporation is a process that requires energy input, and in this case, the energy comes from the surrounding air.
When water molecules evaporate, they transition from the liquid phase to the gas phase. This phase transition requires energy to overcome the intermolecular forces holding the liquid together. The energy required to break these intermolecular forces and convert the water molecules into a gas comes from the surrounding environment.
In the case you described, when water is at thermal equilibrium with the surrounding air, it means that the water and air are at the same temperature. However, individual water molecules still possess a range of kinetic energies due to the distribution of molecular speeds. Some water molecules have enough energy to break away from the liquid surface and become vapor. These higher-energy molecules leave the liquid, causing evaporation.
As the water molecules with higher kinetic energy leave the liquid, they take away energy from the surrounding water, which results in cooling. This cooling effect occurs because the average kinetic energy of the remaining water molecules decreases, reducing the overall temperature of the water.
Therefore, even though the water and air may be at the same temperature, the process of evaporation involves the transfer of energy from the surroundings to the water in order to provide the necessary energy for the phase transition from liquid to gas.