A missionary is someone who has been sent on a mission. Specifically, someone that God has sent on His mission. Because of Jesus’ clear commission sending His people to make disciples of all nations, if you are in Christ, you are a missionary.
The word "missionary" does not appear in scripture. But that doesn't mean it isn't a helpful term. The word is originally latin (all the best English words are transliterated!), and means "sent-one." Related are the concepts of "ambassadors" and "apostles," which should inform our understanding of what it means to be a missionary.
In 2 Corinthians 5:20 Paul writes, "Therefore, we are ambassadors for Christ; certain that God is appealing through us, we plead on Christ's behalf, "Be reconciled to God." An "ambassador is an official representative of a Kingdom to a foreign land with the responsibility (and authority! to speak and act on behalf of the Kingdom he or she represents. Paul's message of reconciliation isn't his own, he is speaking on behalf of Christ, who sent him.
An "apostle" is "one who has been sent to convey a message. As witnesses of Jesus life, instruction, death, and resurrection, the 12 Disciples became Apostles. They were set apart to instruct the early Church in how to understand and apply Jesus' teachings.
The ideas of "ambassador" and "apostle" have be incorporated into the term "missionary." A missionary is one who has been sent on a mission. Passages such as Matthew 28:19-20 make it clear that God has a mission.
Anyone who is in Christ has been sent on God's mission. In John 20:21, we read Jesus' commission to His Disciples- "As the Father has sent me, I also send you." In Acts 1:8, Jesus says, "you will be My witnesses in Jerusalem, in all Judea and Samaria, and to the ends of the earth.”
Missionaries are sent to different people and places, but their role is the same: to translate the universal, unchanging gospel of Jesus into sinful, dynamic human cultures. This requires intentionally crossing cultural (and subcultural) barriers to proclaim the good news in ways that are appropriate to those to whom we've been sent.
An ambassador is an ambassador because he's been appointed to that position. His ambassadorship doesn't depend on his activity, its his identity. Likewise, if you are in Christ, you are a missionary. The question isn't whether or not you've been sent. It's whether you're a good and faithful missionary or a bad and disobedient one.