That's really an interresting topic, I had never hear about yogurt frosting before.
A quick recipe check on the internet shows that it truly consists of yogurt and not much else (if for dogs, and if the frosting should harden). The recipes call for a bit of honey, sugar free peanut butter, etc. as flavor.
Well, doesn't sound bad, but I doubt that it can be used as a 100% substitute for icing. The hardening happens because the yogurt dries, and this will almost always result in cracks (because this is just what yogurt does when left to dry in the air... crack). Additionally yogurt tends to be a bit on the runny side, some decorating techniques require stiffer icing.
I haven't tried it, but how about adding whipped egg white to the icing? As far as I know eggs are OK for dogs. The egg white would give more stability to the whole thing and might also prevent cracks (though not sure about that). Or switch from yogurt to Skyr (special Iclandic thing, kind of a mix between yogurt and quark). The icing would be stiffer to start with (but might still crack...). Or maybe corn starch as a substitute for powdered sugar (could make the cracking worse, though)?
I'd strongly advise you to experiment a bit, sounds like it's worth it 