Is it permitted to give business to someone (or a company) who you have confirmed does not pay income tax (works “off the books”)?
It might be ‘formally’ permitted to do so according to halacha. Paying taxes is the responsibility of the seller, not of the buyer.
At the same time, it is certainly morally correct to prefer taxpaying institutions and sellers who make their just contributions to the State. Buying from people who don’t pay taxes, and even preferring them for cheaper prices, encourages wrongful tax evasion.
The principle question is whether this is a case of mesaye’a lidvar aveirah – abetting a sinful action by giving business to somebody who won’t be paying his taxes.
It is possible that this is not a problem, because the prohibition of mesaye’a involves giving somebody an item with which an aveirah is committed, such as giving somebody a car in the knowledge he will drive on Shabbos. Although this might not be a full case of lifenei iver (say, the car could be otherwise obtained), it remains a case of mesaye’a – aiding and abetting a sinful act. In this case, however, the State does not place of the onus of tax on the buyer, but on the seller alone, so that the buyer does not have to be concerned about it (by contrast with Shabbos transgression).
Mesaye’a is at worst a rabbinic prohibition, and therefore there is generally room for leniency in matters of doubt; please see our article on the topic here for more details.
Yet, the spirit of mesaye’a – the concept of aiding and abetting transgression – certainly applies here, and therefore one should certainly prefer taxpaying institutions.