Well I never had a horse ribbon to wrinkle but I have gotten wrinkles out of ribbons made of all sorts of stuff.
Different materials, that is.
If you think it will melt, try the steamy shower trick. It can't hurt it and might be all you need.
If that doesnt work, do the low heat iron with towel beneath and above as folks here have suggested and keep a close watch on it.
Too bad the folks giving out the ribbons are not more careful!