These can be applied for via the local town hall but they take quite a few days, so you may not get it before your holiday ends!.. if you go to Santa Cruz (weekdays before 1pm) the licence will be issued on the spot. Application form here . Form to pay the fee here, address of the office: Avda. de Anaga, 35, Edf. Usos Múltiples I, 11ª planta 38001, Santa Cruz de Tenerife – Tfno. 922.47.50.00 – Fax 922.24.68.43
if you are in the south of Tenerife there is also a local Fishing club who are happy to help visitors