Q:

Does English contain accented letters? 

Does English contain accented letters?   Is there a standard list of accented letters that are used in the English language? 
In my application I'm currently allowing the characters a-z and áéóúíý äëöüïÿ '-.   The reason I'm allowing '- as well is because some people names contain them. 

A:

You could count accents on imported words (café, resumé, piñata).  Some names also include accents (Zoë, Brontë, Beyoncé).  Then there's the rare case of using accents and umlauts over certain vowels (naïve, coöperation) - this is a non-standard convention not used by the majority of English speakers.    
It's rare to see accents or diacritics in modern English as anything other than a stylistic typographical choice.  A good example of this is Häagen-Dazs ice cream, whose Bronx, NY creators put a superfluous umlaut in the name to add a European flair. 
Many other languages with Latin alphabets use umlauts and other diacritics.  
Examples from the original question are missing ñ, ø, vowels with macron (top bar ā, ī etc) or circumflex (â,ê etc) as well as many others.  
That gets to the key point of the question, which is that you should be asking about the Extended Latin alphabet - not "English."  
Assuming by "application" you mean software, you should look into how Unicode characters work, and what localization options are provided by the platform you're developing for.  This is a complicated topic that goes well beyond just the language, and is better suited to asking at StackOverflow.  

A:

English traditionally uses very few diacritical marks compared to other European languages. 
English employs a huge number of loanwords, however, so to provide enough characters to represent all commonly used English words, you may as well use the entire Latin-1 character set.  We would not want, for example, to deprive people of proper orthography for their

ångströms
arrastãos
daïses
entrepôts
façades
fêtes
Māori
piñatas

Certain romanization systems require still further characters.  For example, the capital of North Korea should technically be rendered as P'yŏngyang and the lingua franca of Malawi is technically Chicheŵa.  

As you can see, where diacritics remain in use, they are mostly retained from their original languages, and are especially retained where they help avoid confusion with another word.  For example, façade is fading in favor of facade, but maté and résumé are still going strong. 
Traditionally, the diaresis (¨) is used to indicate that two adjacent vowels should be pronounced separately instead of as a dipthong, as in naïve.  This usage has been fading fast, with The New Yorker among the few famously clinging to the old convention. 
Otherwise, diacritics are largely found in poetry as scansion markers.  Most commonly, the grave accent (`) indicates that a vowel should be pronounced where it is ordinarily suppressed; for example, blessèd and learnèd have two syllables whereas blessed and learned have one, but such examples are only really found in poetry.  Similarly, the acute accent (´) indicates which syllable should receive stress, the breve (˘) shortens a syllable, and the macron (¯) lengthens it, but these are almost never encountered elsewhere outside of loanwords.