In the Oriental Orthodox tradition, it is similar to what our friend PsaliChrysostom describes above in the Eastern Orthodox Church. The word we use in Arabic in the Coptic Orthodox Church is رسولي (rasouli), which is just the adjective 'apostolic', and is used in all the ways we might use it in English (e.g., الكنيسة الرسولية 'the apostolic church', الإيمان الرسولي 'the apostolic faith', etc.), a potential difference being that we use it with regard to not just those who converted large groups of people (e.g., St. Gregory, the Illuminator of Armenia), but also those who are primarily remembered for their contributions for preserving the apostolic faith, such as HH St. Athanasius, who always gets the honorific 'the Apostolic'/'el rasouli' whenever we mention him. (Though I suppose since he was responsible for sending the Nubian regions their first bishops in the 4th century when he sent the first of them to Philae, it could be argued that HH is apostolic in the other way, too.)
Anyway, so of course there are apostles today, as there are still men and women carrying out the work of evangelizing the world and preserving the faith delivered to us all by the apostles themselves. The key is that they are not one-off or otherwise 'irregulars' relative to Christianity as it is actually established in the world. I could not declare myself an apostle no matter how I read the Bible, for instance, and 'Apostle' is in itself not an established 'rank' within the Christian Church (a point that some parasitic assemblies, such as that of the Mormons, fail to understand). Rather, an apostle of a given time or place will be recognized to be so by the true successors of the original Apostles of Christ, who are the bishops presiding over a given territory and church (e.g., the Georgian Orthodox Church in the country of Georgia, the Armenian Apostolic Church in Armenia, etc.), in the direct line of our fathers and masters the holy apostles and disciples of Christ our God. For example, HH Pope Tawadros II, who has held this position since 2012, is the 118th patriarch to preside over the See of St. Mark, which was established by the apostle and Gospel-writer himself sometime before 68 AD. (As near as I can tell, there has always been at least some degree of debate regarding when exactly our apostle St. Mark arrived in Egypt, but the 68 AD date of his martyrdom does not seem to be contradicted by anyone of note.)