Just because the relay is clicking does not mean it is working. The click is caused when the coil is energized, and the contacts close.
You could be missing the power input at the relay, which is supplied by the horn fuse. I would get a test light on the horn connectors. I doubt both horns quit at the same time.
Looking at the schematic, it is a very simple circuit. At the horn relay you will have two power sources, both supplied by the 10A fuse in the engine room junction box marked HORN.
One power source is for the coil, and the other power source is for the horn relay contacts. When the horn pad is depressed, this supplies a ground for the coil. This causes the contacts to close.
Once the contacts are closed, power flows from the fuse, through the contacts to the horns provided there is power at the contact point and that the circuit from the relay to the horns is intact.
Try hot wiring each horn to verify that they work.
The clockspring in the steering column could be bad, but since you have nothing with the fob too, then I doubt that is the problem.
Looking at my information sources, this is not a common problem with the RAV4.
As a first step, verify with a test light that you have two positions in the relay connector that have power.
