Jesus called his disciples, who are later called apostles.
Mat 4:21 And going on from there he saw two other brothers, James the son of Zebedee and John his brother, in the boat with Zebedee their father, mending their nets, and he called them.
When Judas Iscariot was replaced as apostle, God selected him from the qualified people.
Act 1:24-26 And they prayed and said, “You, Lord, who know the hearts of all, show which one of these two you have chosen (25) to take the place in this ministry and apostleship from which Judas turned aside to go to his own place.” (26) And they cast lots for them, and the lot fell on Matthias, and he was numbered with the eleven apostles.
Paul was called to be an apostle.
Rom 1:1 Paul, a servant of Christ Jesus, called to be an apostle, set apart for the gospel of God,
There is no case in the Bible of an apostle appointing himself except for false apostles.
2Co 11:13 For such men are false apostles, deceitful workmen, disguising themselves as apostles of Christ.
Apostleship was only a calling by God.