The horn button on the steering wheel does not directly supply power to the horn. It just provides a ground path for the coil that operates the horn relay.
One side of the relay's operating coil - the positive side - is always hot from the battery (there is no fuse, circuit protection is provided by a fusible link located on the firewall junction block).
The negative side of the coil is connected to a ground wire that runs to the horn switch located under the steering wheel button. Pressing the horn button closes the switch and completes the ground circuit, the coil is then energized and magnetically closes a set of contacts within the relay housing. When those contacts are "pulled in" they create a path that supplies power to the actual horn(s). The power - that goes through the contacts and subsequently out to the horn - is provided on the same lead that supplies the coil.
When I push the steering wheel button, i can hear a faint clicking sound so I think that the circuit is energized.
You are right on that. It would appear that the entire horn electrical circuit is fully functional. The clicking sound you hear is the coil being energized and the relay pulling in. And since the relay power and the power into the contacts are common, it is a safe assumption that the relay is attempting to operate the horn.
There is just one wire that connects the relay contacts to the horn - that is true even if you are driving a Silverado/Sierra and have the dual horn option. In that case, the power supply to the extra horn is spliced into the primary lead right at the base model horn's connector.
I am not certain of the exact location of the horn(s) on a 1977 C25, but I would be willing to bet that they are mounted to the bottom side of the upper radiator' support frame. Here is an image showing a set of dual horns on a previous truck of mine - just before they were mounted to the radiator frame:
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This shows them as mounted and viewed through the grille:
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Here is how they are fastened to the underside of the radiator support frame:
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Anyway, because the relay is apparently operating correctly, all you need to determine is whether the problem is with the lead that runs from the relay to the horn or - if the diaphragm within the horn itself is seized.
Once you have located your horn it should be easy to make that determination. With an assistant pushing on the horn button - and the wiring disconnected from the horn terminal - test the end of the lead for full 12 VDC. If you observe that power is available to the horn, but the horn doesn't blow...it's seized internally.
Get some Pblaster and squirt a good dose down into the horn throat. After letting the penetrating oil do it's work for a while, bang on the horn a little bit and try it by applying 12 VDC. You will probably notice a pathetic gurgling sound, but that is good. That means the diaphragm inside is beginning to vibrate and the horn will eventually return to normal.
If, on the other hand, there is no power at the end of the lead that feeds the horn...you will either have to find/repair the break in the wiring. Or just run a new length of wire. Save the ends of the original lead and splice the new middle section to them - it will make reconnecting everything much easier.
