Is it permissible for me to recite the adhkar with the intention of protecting someone else with them?

There is no harm in reciting supplications (du'as) and remembrances (adhkar) with the intention of protecting others. This falls under the category of ruqyah (exorcism/spiritual healing) and praying for them in their absence. It is authentically reported that the Prophet (peace be upon him) sought refuge for Hasan and Husayn with the perfect words of Allah from every devil and harmful creature, and from every evil eye. He also used to recite the supplication for pardon and well-being for his family and wealth, and he would perform ruqyah for his companions, and blow over the sick among his family with the Mu'awwidhat (Surahs Al-Falaq and An-Nas).