Vice presidential nominees are normally vetted quite heavily. Part of that vetting process is a review of tax returns to make sure that there is nothing that could be exploited should it get dug up by "opposition research." There was, in fact, a bit of a dust-up in the 1984 campaign over whether or not Geraldine Ferraro's husband had something fishy in his tax returns. For a while, rumor had it that Romney didn't pick Rubio in 2012 because of tax abnormalities. Tax returns do seem to come up in the vice presidential vetting process, maybe as a result of that 1984 thing with Ferraro and her husband.
Not that Trump would ever accept being second fiddle, but let's be blunt. Trump's tax issues alone probably mean that he could never get through a vetting process, based on what we can infer. Funny, that.