WELCOME! See..Still learning something new on BYC. I've never heard of renting. Not sure we have that here in Utah. Glad you are able to have them regardless of health issues. I just gave up my small flock because of certain Heath issues in a few family members. I'm a caregiver to them.. including my husband. Well, I've decided that I should be able to have a couple of laying hens in the summer months. The coop? It's there..to stay. Husband built that years ago. Boy, cement and all.. still standing strong.
Check out Farm & Fleet or Tractor Supply Company. For just 2 hens, they have coops that will be plenty big enough. You can get one with an attached run as well. I think I've seen some around $200.
Also, could you just built one or have somebody like a relative or your husband build one for you? Just some 2x4's, plywood, screws, a few hinges and wire mesh will get a functioning coop built fairly quickly. Not sure if the chicken rental program has specific standards you would need to meet, but DIY can save you money. You may have some of the materials laying around even.
Welcome! Check out some of the small coops members have. Some are repurposed playhouses or small sheds, others made from recycled materials. Check Craigslist or other sites for a used coop or similar structure
It's surprising to me that you have so few predators! I have tons! Including raccoons, skunks, foxes neighborhood dogs, a fox, opossums, and probably more!