" "I can log in on neither the app nor the web" is technically correct, but this is a more formal-sounding sentence, and not something that most people would use in everyday speech. The question mark is not part of the quoted sentence so it should go outside the quotes.
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"either" is correct, but, as you wrote it, the second "on" should be deleted:
"But I still can't log in on either the app or the web."
"But I still can't log in either on the app or on the web."
"I can log in on neither the app nor the web" is technically correct, but this is a more formal-sounding sentence, and not something that most people would use in everyday speec