Is saying, "My Lord, I will not use birth control pills or an IUD if this inflammation goes away," considered a vow? And is it permissible to use them after recovery?

The expression used by the sister who asked the question is not a form of vow (nadhr), as there is nothing in it to indicate commitment. A vow implies the commitment found in contracts. To understand the ruling on using an IUD and the conditions for using contraceptives, please refer to fatwas number 41728 and 39081.