Are there any denominations that believe in contemporary apostles, and if so, how is a person called to be an apostle according to them?

Are there any denominations that believe in contemporary apostles, and if so, how is a person called to be an apostle according to them?

Both the Orthodox and Catholic Churches have historically employed the title of ”Apostle” to various individuals other than Thee 12 Apostles of the Lord.

After all the name apostle simply denotes being sent:

The word "Apostle", from the Greek apostello "to send forth", "to dispatch", has etymologically a very general sense. Apostolos (Apostle) means one who is sent forth, dispatched--in other words, who is entrusted with a mission, rather, a foreign mission. It has, however, a stronger sense than the word messenger, and means as much as a delegate. In the classical writers the word is not frequent. - Apostles (Catholic Encyclopaedia)

However the term apostle has various meanings historically:

Various meanings

It is at once evident that in a Christian sense, everyone who had received a mission from God, or Christ, to man could be called "Apostle". In fact, however, it was reserved to those of the disciples who received this title from Christ. At the same time, like other honourable titles, it was occasionally applied to those who in some way realized the fundamental idea of the name. The word also has various meanings.

• The name Apostle denotes principally one of the twelve disciples who, on a solemn occasion, were called by Christ to a special mission. In the Gospels, however, those disciples are often designated by the expressions of mathetai (the disciples) or dodeka (the Twelve) and, after the treason and death of Judas, even of hendeka (the Eleven). In the Synoptics the name Apostle occurs but seldom with this meaning; only once in Matthew and Mark. But in other books of the New Testament, chiefly in the Epistles of St. Paul and in the Acts, this use of the word is current. Saul of Tarsus, being miraculously converted, and called to preach the Gospel to the heathens, claimed with much insistency this title and its rights.

• In the Epistle to the Hebrews (iii, 1) the name is applied even to Christ, in the original meaning of a delegate sent from God to preach revealed truth to the world.

• The word Apostle has also in the New Testament a larger meaning, and denotes some inferior disciples who, under the direction of the Apostles, preached the Gospel, or contributed to its diffusion; thus Barnabas (Acts 14:4, 14), probably Andronicus and Junias (Romans 16:7), Epaphroditus (Philippians 2:25), two unknown Christians who were delegated for the collection in Corinth (2 Corinthians 8:23). We know not why the honourable name of Apostle is not given to such illustrious missionaries as Timothy, Titus, and others who would equally merit it.

As stated above the title of Apostle has been applied to various other persons in the history of the Church. Here follows a few examples:

• “Mary Magdalene, apostle of the apostles” has been honour with this title by both the Orthodox and Catholic traditions.
• Twelve Apostles of Ireland (also known as Twelve Apostles of Erin) were twelve early Irish monastic saints of the sixth century who studied under St Finian (d. 549) at his famous monastic school Clonard Abbey at Cluain-Eraird (Eraird's Meadow), now Clonard in County Meath.
• Saint Francis Xavier, Apostle of the Indies and Japan

The Catholic Church has employed this title on several persons in recent years including St. Maria Faustyna Kowalska known as the Apostle of Mercy and also as the "Secretary of Divine Mercy".

The Roman Catholic Church canonized Kowalska as a saint on 30 April 2000.3 The mystic is classified in the liturgy as a virgin6 and is venerated within the church as the "Apostle of Divine Mercy". Her tomb is in Divine Sanctuary, Kraków-Łagiewniki, where she spent the end of her life and met confessor Józef Andrasz, who also supported the message of mercy. - Faustina Kowalska

On February 18, 2000 during the Great Jubilee of the Year 2000, Pope John Paul encouraged permanent deacons to become ”apostles of the New Evangelization” It is is as St. Peter himself was sending labourers into the field of evangelization as Christ did with his first Apostles.

Holy Father addresses permanent deacons and their families during Jubilee celebration in Rome

2. In fact, the Jubilee is an important time for self-examination and inner purification, but also for recovering that missionary awareness inherent in the mystery of Christ and the Church. Whoever believes that Christ the Lord is the way, the truth and the life, whoever knows that the Church is his continuation in history, whoever has a personal experience of all this cannot fail, for this very reason, to become fervently missionary. Dear deacons, be active apostles of the new evangelization. Lead everyone to Christ! Through your efforts, may his kingdom also spread in your family, in your workplace, in the parish, in the Diocese, in the whole world!

This mission, at least in intention and zeal, must stir the hearts of sacred ministers and spur them to the total gift of themselves. Let nothing stop you, but persevere in fidelity to Christ, following the example of the deacon Laurence whose revered and celebrated relic you have wished to bring here for this occasion.

In our times too there are people whom God calls to the martyrdom of blood; far more numerous, however, are those believers who must endure the "martyrdom" of misunderstanding. Do not be upset by problems and conflicts but, on the contrary, have ever greater trust in Jesus who redeemed humanity through the martyrdom of the Cross.

Active Apostles of the New Evangelization

I am sure I can find other examples of where the Catholic Church has employed of does employ the title of ”Apostle” on certain individuals. These few are mentioned In order to show that it is a common practice within the Church, but they are not to be considered as equal to the ”Twelve Apostles of the Lord”.

I will give a Lutheran response to the question. It is based upon a response to how some Reformed cessationists hold to what is called a "cascade" theology of spiritual gifts and offices. They argue that if the apostolic office is no longer around than the other gifts and offices should not be either.

The answer to the question depends on what is meant by "apostle" in our current setting. Lutherans would view the canonical apostles (i.e. those who were with Jesus) as a unique group that oversaw & authorized the final formation of the New Testament canon. The gift of primary inspiration (John 14:26), that they received, enabled the church to have a doctrinal foundation. Once the last canonical apostle died, the canon was complete. Lutherans would reject the concept of modern apostles claiming to speak authoritatively apart from the Word of God.

However, Lutherans would also allow for another apostolic category. They would view the gift of apostleship in a broad symbolic missional sense (apostolos from apo = from + stello = send forth).

 For example, in the Lutheran Confessions, the natural reading of "A Treatise on the Power and Primacy of the Pope" (Treatise 26) is that the word "apostle" is used in a broad sense in reference to missional leaders. These missional leaders can be either ordained clergy or lay leaders that are involved in, historically, what has been called the apostolate.

Philip Melanchthon, the author of Treatise 26, preached a memorial sermon to the students at the University of Wittenburg on the death of Luther (1546). He describes this in terms of current church ministry modalities:

The Son of God, as Paul says, sits on the right hand of the Eternal Father, and gives gifts unto men; these gifts are the voice of the Gospel and of the Holy Spirit, with which, as He imparts them, He inspires Prophets, Apostles, Pastors and Teachers, and selects them from this our assembly, that is to say, from those who are yet in the rudiments of divine knowledge, who read, who hear, and who love the prophetic and apostolic writings; nor does he often call to this warfare those who are in the exercise of established power, but it even pleases him to wage war on these very men through leaders chosen from other ranks.

In a similar way, Martin Luther in 1533 writes about the ministry:

For none of us is born as apostle, preacher, teacher, pastor through baptism, but we are all born simply as priests and clerics. Afterward, some are taken from the ranks of such born clerics and called or elected to these which they are to discharge on behalf of all of us. (Luther, The Private Mass and the Consecration of Priests, LW 38:189)

The above comment from Melanchthon is important. That's because many scholars believe that Lutheran Confessional church polity was intentionally set up, among other things, to facilitate an easy "docking" with the Bishop and Church of Rome at some point in the future. It is not insignificant that the Roman tradition still asserts:

The mission of the Church pertains to the salvation of men, which is to be achieved by belief in Christ and by His grace. The apostolate of the Church and of all its members is primarily designed to manifest Christ's message by words and deeds and to communicate His grace to the world. (Documents of Vatican II, Decree on the Apostolate of the Laity ch. 2, No. 6)

One popular Lutheran theologian of the 19th century explains the gift of apostleship this way:

The ordinary ministry (Predigtamt, preaching office) is the continuation, willed by God Himself, of the extraordinary apostolic office, and is in and with the apostolic office of divine institution. (A. Hoenecke, Evangelisch-Lutherische Dogmatik, IV:180); pp. 126,130,135)

In other words, the reference to “apostles” in 1 Corinthians 12:28 is connected to an "appointment” (εθετο) for a "mediated" ministry that parallels what is frequently mentioned as a missional position (e.g. the Sendlinge in the European Lutheran missional tradition). This can also be seen in modern missional categories such as St. Patrick, who is often described as an apostle to Ireland. Another example, might be C.S. Lewis - who is frequently described as an apostle to the skeptics.

The late Greek Orthodox theologian, Archimandrite Eusebius Stephanou, shared this ecumenical thought on the subject:

...we have forgotten the charismatic life of the Church with all the diversity of its gifts. Apart from being ordained to the priesthood, a person can be called to the work of an evangelist and teacher. We can be called to be apostles for missionary work either nearby or in distant places... - Archimandrite Eusebius Stephanou (Rediscovering the Lay Ministry in the Orthodox Church)

The Church of Jesus Christ of Latter-Day Saints has a Quorum of the Twelve Apostles:

https://newsroom.churchofjesuschrist.org/topic/quorum-of-the-twelve-apostles

The modern-day Quorum of the Twelve Apostles was organized in 1835 and is an example of the literal restoration to the earth of the church established by Jesus Christ.

The Quorum of the Twelve Apostles is the second-highest presiding body in the government of the Church. Its members serve under the direction of the First Presidency, a governing unit of three men — the president and two counselors.

In addition to their primary responsibility to be special witnesses of the name of Christ throughout the world, the apostles have heavy administrative responsibilities as they oversee the orderly progress and development of the global Church.

Just as their counterparts in ancient times were sent throughout the world, the apostles today travel the world to strengthen and encourage Church members, to organize new congregations and to conduct the business of the Church. Sometimes this means meeting with national leaders to negotiate permission for the Church to be established in yet another country.

A replacement in the Quorum of the Twelve is called by the President of the Church, who seeks and receives inspiration in extending the call. The new member of the Twelve may be called from the general authorities who are senior leaders in the Church or from the general Church membership around the world. Seniority in the Quorum of the Twelve is determined by the date an apostle is called rather than by age.