By now you are probably back home, and I wonder if (let's say I doubt) you found what you were looking for. The Costa Blanca doesn't appear primarily as a place favored by cyclists, the most common kinds of sports there being more like sunbathing and beer drinking. It is a pity, as there are many nice small roads in the back country. The sea front itself is mostly to be avoided. Too much incompetent traffic, and too much traffic overall.
Back to renting a bike: the only real - but small - bike shop I know of is out of Moraira, left hand on the road toward Calpe. In principle there should however be more in Gandia or Denia, as these are supposed to be real towns and not just mere resorts.
The Costa Blanca is quite a big place and is not as bad as caracol40 makes out. Once you are off the main roads it is perfect riding conditions. Even on the main roads there are cycle paths. Javea and Denia are the best destinations with flat and mountainous roads. There is a bike shop in the old port of Javea.