Does time exist in a vacuum?

Of course, time exists in a vacuum. In fact, time cannot be considered in isolation. Einstein showed that the proper coordinates for any event includes length, width, height, and time, or x, y, z, t, which is equivalent to i, j, k, t. This is called "space-time" and, as you can see, it's four-dimensional.

Also, a vacuum in NOT merely "empty space." Even in intergalactic space, there are random atoms floating around. Their density might be only, say, 1 atom per cubic meter, but that adds up over the entire span of the known universe.

Furthermore, there are what are known as virtual particles. Matter-antimatter pairs come into existence then proceed to annihilate each other, the duration (time) of their existence being inversely proportional to their mass.

Earth is surrounded by vacuum. Aren't you aging?

Vacuum is just a scarcity of atoms. Does not mean that time ceases to exist.

Time doesn't exist. It's an artificial construct created by humans to measure the decay of particles based upon the rotation cycle of celestial bodies. The only time that does exist is right here and right now.

Imagine you are watching a video, but the video is black, nothing happens, ever. So as you are watching this video you look at your watch. time moves normally. My point being, if you were in the video, nothing happens, nothing changes, times might as well not exist. But to the outside observer time never stops.