You can replace the brass ring from the dealer. I am almost sure there is no relay. If you can take a wrench and touch the steering shaft and the little ring around the steering shaft and it beeps, it is in the button and the spring and brass ring. I use a .22 shell over the spring. I have changed the spring but it takes some finesse as there isn't a whole lot of room.
Below is copied from my thread.
Plate horn contact part # 84312-20011. $4.98 through an online Toyota discount parts dealer.
Here is a better picture of the Brass Ring and my pointer finger is what makes contact to sound the horn. I just clean them with some fine grit sandpaper.
Got the horn to working. I used a .22 shell to make the contact to the steering wheel. I found this on Yotatech and like it better then how I used to fix my horns. I have it listed on my 88 4Runners build thread for more details about getting the horn to work on the first page.
Another thing I got fixed on my 91 was the horn. I know this happens alot and hopefully this will help some in the future. Pull the steering wheel with a Steering Wheel Puller that you can usually rent from a parts store for free. Then check it by touching the brass spring in the 11 oclock postion. I just used my wrench. If it honks it is most likely the brass ring on the back side of the wheel. Not the best picture but you should get the idea.