It depends on your definition.
Does the specific definition of apostles that are described in the New Testament exist today? No.
Are there special messengers for the LORD’s message today, that the word “apostle” could be? Yes. We functionally call those messengers, missionaries.
In recent years there has been a lot of interest in studying Ephesians 4:11-12 which states:
And He personally gave some to be apostles, some prophets, some evangelists, some pastors and teachers, for the training of the saints in the work of ministry, to build up the body of Christ
I have gone back and forth some in my working definitions of the term apostle, so I give you my current opinion with caution. Yet, when faced with theological ambiguity, its probably best to choose a conservative position, which I currently take.
Much of the debate over the role of an apostle today is differences in nomenclature, and not a reason for further division in the church.
In order to be an apostle in the Bible, you must have:
a. Seen Christ physically (Acts 1:22)
b. Personally been appointed by Christ to be an apostle (Matthew 10:1-7, Acts 9:5-6, 1 Corinthians 9:1, 15:7-9)
c. The apostles’ letters were authorized as Scripture (with the exception of Mark, Luke, Acts, Hebrews and Jude – who were written by men close to the apostles, and apparently authorized by the apostles.).
We know that there were twelve original apostles appointed by Christ (the eleven plus Matthias), Barnabas, and Paul (Acts 14:14). The original twelve are so important an unique that their names are inscribed on the foundations of the New Jerusalem (Revelation 21:14).
There were likely other apostles, such as James the brother of Jesus, but there seems to be no apostles appointed after Paul.
2. It’s not sinful to use the term “apostle” – just not helpful
There are powerful, Biblical ministers who call themselves an apostle. While I don’t think using the term today is accurate or beneficial, it certainly wouldn’t be the exclusive reason for me to not receive from those who call themselves an apostle. Again, it can simply be a disagreement on terminology.
3. If you need the term “apostle” to be a functioning term, an apostle is a “missionary”
Some people simply cannot accept that a Biblical office is no longer functioning. For those who need a definition for the word apostle, it is good to refer them as a missionary. A general meaning of the word apostle is the term messenger. This is used three times in the New Testament (Philippians 2:25, 2 Corinthians 8:23, John 13:16).
Grudem, W. A. (1994). Systematic theology: An introduction to biblical doctrine. Leicester, England: Inter-Varsity Press.