I lost my identity card a few days ago and need to have a new one issued. What is the Italian verb for "issue" in this case?
The proper translation of the verb "to issue" is "rilasciare", so you would say:
"Devo farmi rilasciare una nuova carta d'identità".
However, commonly you would use the active form, saying, for example:
"Devo richiedere una nuova carta di identità" (formal)
"Devo rifare la carta di identità" (informal and normally used while speaking)
"Rifare" is a verb which is mostly used in informal contexts and which means "fare di nuovo" (to do again).