Yes, the Bible clearly establishes that Paul was an apostle.
Romans 1:1Paul, a servant of Christ Jesus, called to be an apostle and set apart for the gospel of God–
Romans 11:13I am talking to you Gentiles. Inasmuch as I am the apostle to the Gentiles, I make much of my ministry 14in the hope that I may somehow arouse my own people to envy and save some of them.
1 Corinthians 9:1Am I not free? Am I not an apostle? Have I not seen Jesus our Lord? Are you not the result of my work in the Lord? 2Even though I may not be an apostle to others, surely I am to you! For you are the seal of my apostleship in the Lord.
1 Corinthians 15:9For I am the least of the apostles and do not even deserve to be called an apostle, because I persecuted the church of God. 10But by the grace of God I am what I am, and his grace to me was not without effect. No, I worked harder than all of them–yet not I, but the grace of God that was with me.
2 Corinthians 12:11I have made a fool of myself, but you drove me to it. I ought to have been commended by you, for I am not in the least inferior to the “superapostles,” even though I am nothing. 12The things that mark an apostle–signs, wonders and miracles–were done among you with great perseverance.
Galatians 2:8For God, who was at work in the ministry of Peter as an apostle to the Jews, was also at work in my ministry as an apostle to the Gentiles. 9James, Peter and John, those reputed to be pillars, gave me and Barnabas the right hand of fellowship when they recognized the grace given to me.