Cindy.
A couple of years ago, I had a small infestation of cat fleas from our late cat.
Like you, I don't like these chemicals at all and won't use them. But I did a load of research and discovered that, apparently, most insects have a very keen sense of smell and hate some smells.
It a matter of experimentation really. The strong smells like garlic are obvious. But others that work with many insects include, Rosemary, Catnip, Basil and many others. The problem is, each type of insect seems to be sensitive to some more than others, so you absolutely have to experiment.
It's best to get oil essence of these plants and dilute it, a few drops in something like surgical spirit. The put in a sprayer. Never put oil essence directly onto skin. Once diluted, it can be safely sprayed onto much of you. Including your hair!
It certainly worked on those fleas. As a repellent to stop them biting. I got rid of them by vacuming. It seems that fleas like most insects are actually creatures of habit. They tend to stick to certain areas. Fleas like to sit in shadowy areas, such as the edge between furniture and carpets. They do like carpets.
I have absolutely no doubt that mosquitoes will be basically the same. A bit more mobile and actually getting rid of them is impossible. But they can be repelled with smell.