Yes, fishing from the sand is allowed, provided you have the appropriate fishing license.
During the swimming season (from the Friday before Easter to the first Sunday in November), you can fish from 8pm to 8am the following day. In the section between Punta del Cavall and Pas de la Mala Dona St. (Cap Salou) fishing is allowed 24 hours a day all year round.
* Exception: it is totally forbidden to fish in Cala dels Crancs and the adjacent areas, you cannot fish on any day or any time of the year, due to the protection of a seriously threatened species.