I get from my payment processor a token in place of the cardholder's CC number. To make the payment I have to ask the cardholder for his CVV. Doing this by calling the payment processor's webpage is cumbersome. Is it allowed to ask for the CVV in an app and forward that with the card-token to the payment processor? I think it could be allowed as I am not in possession of the actual card number and I am of course not going to store the CVV.
No. The CVV is "Sensitive Authentication Data", and according to the PCI DSS page 7:
PCI DSS also applies to all other entities that store, process, or transmit cardholder data and/or sensitive authentication data.
So, yes, transmitting (receiving and re-sending) just the CVV is enough to put you in scope; if you're already using Tokenization, you're probably doing that to stay out of scope, and this would be a bad idea for you.