Gail
I have seen "rental livestock" live and in the news, both around Chicago and in CA. But I believe it's very rare. Since I have personally seen "contract" sheep and goats as well as dogs, I know people do that. But I have never come across working animals more than once or twice every 5+ years. Not common at all and that makes me think it requires a home base and a particular defined geographic area.
You're asking, in essence, about a business plan, market saturation, operational realities - that kind of hard core nuts/bolts. I think. Perhaps the best approach might be to treat it like business research and try to ferret out existing contractors and see if they would share. One approach would be to imagine yourself as a customer and try to follow those channels. But big commercial ranches may not be helpful. Their biz model is heavy and complex
enough that incorporating a small scale contract operation would just not be worth it to them. Ag schools might be one place people would have heard or know. Ads in the
local craigslist or campus forums might be worth a try.
You might try to contacting Travis Johnson here by PM for his thoughts. He has strong opinions, but he's done a lot of farming (and everything else). He's not a big corporation and he made sheep work for him for many years. I mean work as a paying business, starting from not much. Now that's not directly to your question, but... He thinks out of the box and generally just has a lot
experience getting things done. But I haven't seen him posting lately so it may be that he's gotten real busy with "stuff". Worth "talking" to if you can reach him, though.
Another idea might be to contact golf courses or other likely entities which could have been customers. For some reason I suspect urban areas may be easier to find something...
I think you need to talk with people who have/are doing that business if you hope to get good info. It's possible you may need to cast a very wide net, but, at least 6-7 years ago there were rental sheep around Chicago (or Lake Geneva - I really can't remember where I saw them) so I'm guessing there are some people doing it somewhere. If you find a sheep person, they might have info on a goat person.
Or maybe somebody here's going to chime in with just what you need! But I think it will help _greatly_ to approach it as a business proposition in order to give yourself the hard headed attitude that the operation will require to succeed.
Best luck,
Rufus