On our trip last summer, the first 10 days of the trip was camping. Most campsites in Iceland do not even accept reservations, and the ones that do are the ones closest to Reykjavik. Even in that area I would not worry as there as so many campgrounds that if one were full you could just move on to the next one.
As for tips, here are a a few. Some places will charge for showers, so you will want some 100kr coins. Bring shower shoes!!! Also, bring a bag for all of your shower items that can either be hung up or if you have to set it on a floor, it won't get all gross. Some showers have a chair to put stuff on, or hooks, but others are very bare bones. A number of campsites have washers and dryers, although maybe only one machine of each. Those you will usually need the 100kr coins as well, and each wash and each dry will run around 600-900kr. They typically do not provide detergent, so bring laundry sheets with you. The campgrounds are just a big area of grassy land, not specific camp sites like in the US. Just park wherever you want, but be sensitive to how close you park to the next person. Give a bit of space. Many campgrounds have communal kitchens, but we only used those once when weather was bad. Otherwise we just cooked in the camper or on a grill that we had with us, but if you can buy/rent/bring a campstove that will make it much easier. When you arrive, the host will usually come find you to pay for your spot. All of the ones we stayed at took cards. If the host does not find you, look for them in the morning. Sometimes there is a way to pay via the Parka app, so just do that upon arrival.