Must the master of an offshore supply vessel arriving at a UK port from a North Sea rig located on the UK continental shelf report to Customs?

Yes.

The UK continental shelf (UKCS) is treated by HM Revenue and Customs as if it were outside the EU. Supply vessels servicing installations on the UKCS from UK ports may therefore be required to report and clear as if they were vessels arriving from or departing for non-EU ports which are not on an “authorized regular shipping service”.