Firstly, keep in mind that there's no live virus in the vaccine, depending on the vaccine it's either a protein from the virus or a RNA molecule encoding a protein of the virus, neither of which can replicate in the human body. So the most there ever will exist is in that person's arm immediately after the shot, then it slowly gets cleared from the body over time.
So while it's theoretically possible (though unlikely) that a few of those molecules make it through the bloodstream of the vaccine recipient and into his or her digestive secretions, bile, etc. and get excreted in the feces, the concentration will be FAR lower than it is in the injection site in the person's arm. Our mucous membranes are exposed to foreign proteins from all sorts of organisms all the time, having a few coronavirus proteins touch them is not any different (and if you've been on Earth since 2020, you likely have had single particles of this virus somewhere on your body at some time).
I wouldn't worry about it too much even if the person was vaccinated yesterday--and if the vaccination was months ago then it's as if it never happened. I mean for the purposes of "second hand vaccination" like you seem to be implying--a vaccinated person is less likely to have gotten sick with COVID recently, and being sick with COVID may well have an impact on the microbiome, so in THAT sense it could possibly make a real difference, indirectly.
