Both are correct everywhere.

courriel is preferred where it is appreciated to use the French words for things, like in Québec, or when exchanging with a lawyer, but in other situations when such strictness is not required, mail and email are more natural.

Also note that the use of courriel increases with age – I have yet to see someone under 20 use this word in a casual conversation!

Sure, but why would you? Who are you writing this to? The term courriel (GDT, Wiktionnaire ; see also the links on Ac.9) is recommended in Québec and by the Commission d’enrichissement de la langue française in France. That being said usage varies depending on region. In part because the relationship with English language loanwords may vary from a country to another (or even from a person to the next). Which in turn is the case because the sociolinguistic makeup is not the same everywhere.¹

In France, chances are mail is quite typical and the Wiktionnaire entry might sum it up correctly in e-mail : « (Anglicisme informatique) (Internet) Variante de mail, plus fréquent, et de courriel, recommandé par l’administration » ; it also says mail is informal. In Québec I’ll sometimes use courriel or message, depending on context. You should select the term based on usage in context (region; formal, casual); native speakers should understand any of those words irrespective of personal usage or preference.

As for par retour de courrier, it essentially means as soon as you receive this, without delay (Larousse en ligne, Wiktionnaire). Sometimes when this construction is used with (e-)mail/courriel it implies you can forward your documents electronically i.e. files attached as opposed to actually mailing them, but without any sense of urgency. Formal communication shouldn’t contain obscure formal clauses casting doubt on the expected course of action. So if you want to specify the moment, delay or method for transferring documents, IMHO you should simply say so and be done with that turn of phrase altogether:

Merci de m’envoyer/me faire parvenir les documents demandés par courriel/(e-)mail (dans les plus brefs délais [as soon as possible]/sans délai [with no delay]/avant le 18 janvier [before January 18] etc.).

_{¹ To make a very long story short (1, 2, 3), and somewhat simplistic I’m afraid, Québec borrowed more English language words earlier on, because of proximity (to the U.S.) and history (the British rule), and does less so now whereas in France it’s arguably happening more so now in comparison, especially with the technology related lexicon. See the anglicismes, emprunts, anglais and traduction tags for content on some of those etc.}