What can I do for a stray dog's cough?

Not letting you touch the ears usually would lead me to think that there could be an ear infection. Except that Canine Influenza is a current (possibly fatal) threat. A couple months ago, an aggressive strain of Canine influenza showed up in my area, such that groomers and vets in my area started requiring dogs to be vaccinated in order to receive service and vets treat dogs who show any such symptoms as if positive.

Since you indicated that going to a vet is not a possibility, I would suggest reaching out to some local rescues near you and seeing if any of them use a traveling vet that helps their dogs awaiting homes (& if they could refer). Rescue groups often get deals on veterinarian care.

I rescued a puppy 9 years ago. Never put him on heartworm meds because his tests were negative. 3 yrs ago he turned up heatworm+ after I took him to Vet for bad cough that wouldn’t go away. I guess an infected mosquito bit him. It can be a symptom of that. I had assumed that It was from choking so hard on walks from his collar (he had never been handled by people). I ended up buying a harness instead but he never fully got over whatever damage had occurred I guess. The combination of both issues. A big lesson learned on my part. Some dogs have irritated throats due to many reasons and it’s treatable w/Veterinarian help. I got a prescription cough medicine (100/month) to help my dog. I hated hearing him gag constantly. It broke my heart. There’s nothing really OTC that helps except for what I use now. Expensive but worth it for them to have good quality of life in my opinion.

Your dog needs to be seen by the Vet. Once you get her to the Vet's office, if the dog will not allow the Vet to touch her, the Vet can prescribe a mild sedative so she can be examined.

But getting her checked us your best option. There are so many different things that can cause a cough. Ranging between Kennel Cough, a piece of grass stuck at the back of her throat, to something more scary like Heart Worm. (And no , you can not but Heartworm meds over the counter. The dogs need to be tested first to make sure negative prior to starting medication. ) Sorry, wish I could be more helpful, but there are so many different reasons dogs cough.
Hopefully you get her to the vet soon!

This is a snowball of an answer. I've worked with ferals for about 10 years.

Look into Julie Hart in NM, she's pretty close to you being in AZ. I'm in St. George, UT and would also love to help with your rehabilitation. I've done it hundreds of times.

As far as a the cough, I wouldn't worry too much about it if it isn't severe. This is a street dog and doesn't have an owner though?

You are limited through ownership laws, as animals are treated like property. I'm inclined to say find the owner, or get the animal shelter involved and get her taken in somewhere where she can get into a rescue to get care.