In this context, "Prüfen" is possible but uncommon. It means more like "to inspect". More common ways to say it are:
- "E-Mails lesen" (read mail) (as Em1 pointed out)
- "E-Mails abrufen/abholen" (fetch mail)
- "Nach E-Mails sehen/gucken/schauen" (look after mail)
- "E-Mails checken" ("checken" is a colloquialism commonly used in youth language)
Note that unlike in English, German people say "E-Mails" when using the plural form and "E-Mail" when referring to a single mail. Additionally, "E-Mail" is often abbreviated with "Mail" in spoken and informal language.
"Prüfen" or "überprüfen" is more common when you want to say "check your inbox". "Prüfen" or "überprüfen" in this context is more closely defined as "to validate" or "to verify":
- "Das Postfach prüfen/überprüfen" (check the mailbox/inbox)
- "Ins Postfach schauen" (look into your inbox)
This variant is used in more formal contexts, when dealing with customers, in business language or in manuals. For example, if you order something at an online store and they want you to check your mail, they'd say: "Bitte überprüfen Sie Ihr Postfach" (please check your inbox).