I was doing research, and I can't seem to find proof that the Portuguese used privateers.
I think I'm a bit lost in translation, because I'm not sure what you mean by privateers.
Do you mean privately owned ships? If so, yes, of course. The owners were called armadores and they belonged to a partnership, called Companhia das Naus, that worked like a type of insurance company for ship owners and was essential for the Age of Exploration.
If you mean private explorers, yes, as well. They usually had to pay the king for the right of exploring and trade monopoly and also give the crown part of the lands they explored. One good example was Fernão Gomes da Mina: King Afonso V gave him the monopoly of exploration and trade of the Golf of Guinea for five years (later six), in exchange of 200 reais yearly, the requirement of exploring at least 100 leagues every year and giving the crown 100 leagues at the end of his contract. Later he went on to sponsor many other private explorers.
http://cctic.ese.ipsantarem.pt/red/hist/ficha_documento.php?cod=84