Basically you can use anything that's a positive experience for the horse as positive reinforcement. It just needs to be really rewarding. In some situations even food is not rewarding...
The problem with alternative rewards is usually the timely application. Some horses love scratches, but they take a long time and a calm horse is needed to appreciate it
It 's hard to scratch a horse that's about to experiment with rears
But often just being interesting and playful is very rewarding.
If you want to clicker train some of the more complex things or want to micro-shape a particular movement it might be hard to do without food rewards that you can administer at a higher rate.
But I'm pretty sure that there is some food you can give your horse - even with Cushing syndrome. I'm reading that carrots are OK also for horses with laminitis and they can be chopped into very small pieces, so that one carrot makes for a good clicker session. Also like you said, pelleted hay is surely an option.
Sometimes I also use the daily ration of minerals as clicker treats - my horse loves that