Pineteam
There is absolutely no restrictions to use vertical speed during climb by Airbus.
There's VS mode so obviously it can be used. But what is not restricted doesn't automatically become recommended. CLB and OP CLB are recommended modes speed can be managed or selected. Nowadays VS is generally used in RVSM towards last few thousand to prevent TCAS. Incidentally there is an Airbus mod called TCAP (TCAS Prevention) which automatically reduces the VS to 1000ft on receiving a TA.