Follow these steps

Seek medical attention immediately.
If you're the one who is injured, someone who is supervising you or your travelling companion will attend to this. Most licensed supervisor's will be trained to critically assess you and help you get the right treatment if you're injured .

Get in touch with your travel insurance provider.
If you're unable to do this yourself, have someone contact them on your behalf. Your insurer can help you find hospitals overseas and even help you guarantee emergency medical costs up-front e.g. with ambulances and emergency surgeries.

Compile all medical documents and incident reports.
This will help you make a claim for any medical costs or for the cost of additional travel arrangements that come out of your own pocket. Remember to get the insurance details of the licensed horse riding provider too.

Make a claim.
Fill out and submit a claim form from your insurance provider along with any necessary documents.