According to Jastrow, גמילות in the expression גמילות חסד is a noun, not a verb. The singular of the noun is גמול or גמולא which means deed, reward or recompense.
It is worth noting that Jastrow says it is Aramaic. The Hebrew form is גמל. And the Hebrew form as a noun is associated with both the camel (the one who carries the physical load) and the camel driver (the one driving the animal/camel, the Nefesh HaBehamit). So in context, it would have a meaning of an executor, one who executes the action.
Another interesting point from Jastrow concerning the usage as it refers to the camel driver, is that the camel driver leads his animal (walks in front of it), in contrast to an ass driver (חמר), who follows behind the animal.
The verb form in Hebrew also appears in Bereshit 21:8 and according to the Targum has a connotation of finishing or completing. This would mean that Gemilut Chesed is bringing Kindness from potential into an actual, finished state.