There is no theological reason why it is impossible to see Shaytan, although most people do not see Shaytan on a regular basis.
However, how does your friend know it is Shaytan, specifically? Is he arriving with an ID card? What about just garden-variety shayateen?
Basically there is no way to prove things which are not objectively measurable. That is, there is no way to prove he is seeing Shaytan and there is no way to prove that he is not. All you can say is that he sees something, and this is what he understands it to be.
It is said that doing wrong acts leaves an ugly imprint on the soul which becomes manifest in the hereafter. For this reason, some mystics are said to be able to see people's true natures. So perhaps this might also be what he is seeing.
I might be suspicious of the clause "whenever someone does something wrong according to Islam", because sometimes people do things that are ethically wrong but are not obviously forbidden. For instance, it is allowed to buy a knife. However, if I buy a knife to murder someone, this is very wrong. Still, it would not be obvious from my purchase that I am committing a sin.
Similarly, we do not always know when someone is doing something wrong. For instance, someone might be lying or trying to flatter someone in order to take advantage of them, but we do not always know it. Along the same lines, we don't usually know if someone is secretly breaking their fast or committing other hidden sins.
(Of course if he is seeing Shaytan in these cases, it is a different story.)
Anyway, most people do not require Shaytan to do wrong things as we are quite capable of doing them ourselves without him.