On your clothes/skin, yes. However, if the virus is unable to replicate in your system, then you're not going to be able to spread it.
Essentially, vaccines provide a sample of whatever it is (be it virus or bacteria) to your body so that it can develop antibodies to it. If you get exposed to the item that you're vaccinated against, the white blood cells will immediately attack the item, and usually, you won't get sick since the item won't reproduce enough to be able to be spread.
In many cases, a person is contagious before they even show signs of being sick. The symptoms of being sick simply means that the virus/bacteria has replicated enough to cause issues and the body is fighting back.