Is is reasonable to expect to get sleeping and eating accommodations in Galway and/or Dingle Town, without a reservation during mid-September?
I would say not. At least not in Dingle. We made our reservations for early September a couple of months ago. I tried changing a few of our reservations (not Dingle or Galway but Dublin and Kinsale) and could not find anything available. I would certainly check it out for your dates now and see what availability is.
We stayed at a Galway B&B in September 2019, and they were almost full, but not totally booked. But that was 4 years ago. There are enough hotels that you’ll get a room somehow, but you may need to look around once you reach the city, and you may or may not pay more for the available accommodations. If you have your eyes set on a particular place, book early. If you’re not so picky, you’ll likely be able to find a place.
As noted above, Dingle will be harder to find a place on the fly.
I would imagine that it’s probably busier right now than it will be in September. Using booking.com, look at what is available tomorrow night. If there are plenty of options in your price range and to your taste, it’s probably a good indicator of what you can expect in September. Unless there are local festivals happening, for example.
I no longer travel without reservations for lodging. We're not that picky about food, so I rarely get a reservation, although last May I did get one in Dingle for the seafood place.
If you know you'll be in a certain place, what is the harm in making a reservation? Most are cancellable a few days before, although you should check the terms. The internet and sites like booking.com have created a situation like a nuclear arms race, where folks are getting their reservations earlier and earlier with less and less of commitment to eventually staying in that spot. If you look one year ahead, you'll find many places are filling up. Others don't open their availability till Jan 1 (for example) and you see a surge in availability. I surrendered and buy my flights 11 months in advance and usually have my lodging booked within a few weeks of that.
If you're a party of one or two, finding lodging is more likely, but a larger family or group shouldn't count on it, at least not during the busy season.