First, it's important to note that before answering the question, "Are there apostles today?" we need to answer the question, "What is an apostle?" which in turn implies, "What was an apostle in the first century?" And that's not as easy a question to answer as one might think, because different NT writers used the term differently.
Most of the time that Luke uses the term apostle, he restricts his meaning to the Twelve (as in Luke 6:13). (Beginning in Acts 1 these are the Twelve minus Judas Iscariot plus Matthias.) Paul is much more flexible in his use of apostle. First, he repeatedly calls himself an apostle of the same rank as the Twelve because he too saw the risen Jesus and received a direct commission from him (see 1 Cor. 15:8-10). Second, Paul uses the term apostle to apply not just to the Twelve but to all the elders in the Jerusalem mother-church (note James in Gal. 1:19 and "all the apostles" in 1 Cor. 15:7). And third, Paul uses the term apostle to refer to any Christian sent out by a church on a mission, whatever it was (note 2 Cor. 8:23; Phil. 2:25). Perhaps it is on this basis that Paul implies that Apollos is an apostle (1 Cor. 4:6, 9).
So, are there apostles today? If by apostles you mean the Twelve, or the Twelve plus Paul, or all the elders of the Jerusalem church, then yes, but they are united with us in heaven and not on this earth. (Though perhaps you could argue that current priests and pastors in Jerusalem today are apostles by the last of those definitions.)
But if an apostle is any Christian sent out by a church on a mission, then there must indeed be apostles today. Every missionary is an apostle. Perhaps traveling shepherds of churches like judicatory clergy (district superintendent, etc.) are apostles. A church planter would be an apostle too, I think. Delegates from churches to consultative bodies might be considered apostles. People who bring financial or physical relief to afflicted fellow-Christians serve as apostles, at least while they're doing it.
At least one key in defining the term apostle is that an apostle is sent out to move beyond a home base, perhaps to return or perhaps not. Apostles aren't wanderers, but they are travelers. Wherever they're staying is just a temporary holdover in their process of going, and that's obvious to everyone.