Flagging mechanisms on social media platforms allow users to report inappropriate posts/accounts for review by content moderators. These reports are pivotal to platforms' efforts toward regulating norm violations. This paper examines how platforms' design choices in implementing flagging mechanisms influence flaggers' perceptions of content moderation. We conducted a survey experiment asking US respondents (N=2,936) to flag inappropriate posts using one of 54 randomly assigned flagging implementations. After flagging, participants rated their fairness perceptions of the flag submission process along the dimensions of consistency, transparency, and voice (agency). We found that participants perceived greater transparency when flagging interfaces included community guidelines and greater voice when they incorporated a text box for open-ended feedback. Our qualitative analysis highlights user needs for improved accessibility, educational support for reporting, and protections against false flags. We offer design recommendations for building fairer flagging systems without exacerbating the cognitive burden of submitting flags.