This depends on how apostles are defined. Biblically, the word "apostle" literally means "one who is sent." So, anyone who was sent by the Lord to do something would be an apostle. But, such a liberal definition is not a sufficient answer. Biblically, an apostle was someone who was involved with Jesus and/or knew of Jesus before His crucifixion and after His crucifixion. Consider the following two sets of verses.
- Acts 1:21-26, “It is therefore necessary that of the men who have accompanied us all the time that the Lord Jesus went in and out among us--22 beginning with the baptism of John, until the day that He was taken up from us--one of these should become a witness with us of His resurrection. 23 And they put forward two men, Joseph called Barsabbas (who was also called Justus), and Matthias. 24 And they prayed, and said, You, Lord, who know the hearts of all men, show which one of these two You have chosen 25 to occupy this ministry and apostleship from which Judas turned aside to go to his own place. 26 And they drew lots for them, and the lot fell to Matthias; and he was numbered with the eleven apostles.”
- 1 Cor. 9:1, “Am I not free? Am I not an apostle? Have I not seen Jesus our Lord? Are you not my work in the Lord?”
Notice that in Acts when the apostles are deciding on a replacement for Judas, Peter speaks of the necessity of someone who had been with Christ from the beginning. In 1 Corinthians Paul defends his apostleship by claiming to have seen the risen Lord. Therefore, we can conclude Biblically that a true apostle in the New Testament style is no longer possible because it would require that the person had been with Christ and/or have seen the risen Lord.