all you can do is wake your horses. I have in my forum the full list of objectives so you can see what is coming next and prepare for them.

You can get them as a gift from friends or from certain players' UFOs on their pages. You can win an aging point from Archimedes divines (look in the directory and check the Special Horses box), which will give you another chance to get droppings from your horses. You can put the items you're looking for on your presentation page, since unsolicited begging for items via PM is against the rules.

Other than this, you'll have to wait for your horses to age for droppings since they cannot be bought in the store.

Don't put Pluto's Parchment on your horses until you have a little more droppings in stock (it gives you equus, but the horse will no longer produce droppings).

Foals do not produce droppings until they are over 6 months old. Horses that eat more/are naturally heavier (large riding breeds and Drafts) will produce more than horses that are lighter weight (smaller riding breeds, ponies, foals.) Horses that are worked throughout the day can eat more (and thus produce more droppings) without putting on too much weight and losing health, so working your horses is recommended.