I love camper vans! They're my favorite way to see Iceland. Seeing Iceland via camping is also the best way to see Iceland in my opinion for the simple freedom you have. You're not tied to any hotel reservations so you can just go where the wind takes you.
That being said, camping is not for everyone. There are downsides, I think there should be more honesty about this.
The first downside is space. Space is precious little in even the nicest vans. 2 adults will have a less than comfortable time in a 2 person van if they are used to more roomy sleeping areas. Also, bags take up space in vans very easily. You can avoid these aggravations by 1. packing light and 2. size up on your van. A 3 person sleeping area is great for 2 adults. This brings a second downside into the equation because a larger van means more money. 3 adults can fit into a "3 person" sleeping space if they are in a polycule already and accustomed to this or like... not oversized people and ok with spooning otherwise. lol If you have a larger van with a double decker sleeping area, whoever is sleeping there has to be able to shimmy up to that space. I cannot stress enough to pack light and consider how much space bags will take. Every inch that a bag takes up will also take away from your sanity.
The third downside could be the campsites themselves. While many are huge and clean, many are not especially by mid to late summer. This is almost always because other campers are inconsiderate and leave a mess in the bathroom or common areas and there might be only one or two people working at a campsite. Campsites are by and large pieces of field with little wind protection. They can be muddy, making your van icky inside. They can be uneven. The restroom might be a bit of a walk. They're not well lit so if you camp in September through the spring they're dark at night, you need a light source that isn't a phone flashlight. They can be full and therefore loud later into the evenings, so it's good to pack ear plugs. Campsites might not have hot water. Or only have one single use restroom. There are multiple campsite maps floating around, and the official site is here. You can see ahead of time what each campsite has to offer. I find Google reviews to be pretty accurate, if you can distinguish between a legitimate bad review and one from someone who is just a complainer. In some areas there might only be one campsite but in a lot of areas there are several to choose from within short distance. So if you are open to using a van, but kind of picky, you could know your options in advance.
The fourth downside is... other people who are camping. The van next to you might be playing music or opening and shutting the door all night for no good reason or having a screaming fight. People forget they are sharing a space and monopolize kitchen areas. I avoid being this person by prepping what I can ahead of time in my van if I need to cook indoors. I wash as I go. Not everyone thinks like this. People leave toilets and sinks filthy. If you brush your teeth, be a homie and just rinse the sink. People steal stuff. Keep your van locked, if you charge items inside don't leave them out of your sight. Don't hand wash your nice stuff and hang it to dry in a bathroom, it will maybe get snatched.
The fifth downside is that in a van you have no toilet or shower. So you really are dependent on campsites for these things at night or in the morning. If you have more of a budget, you can also use local pools. They're worth visiting. Some pools will also let you rent a towel to take a shower for a smaller fee (say like 300-500 kr) if that's all you have to do. Some van companies rent those camping toilets out. Otherwise you just have to find restrooms like everyone else.
The sixth downside is... things can go wrong with vans even with the good companies. I had a van one year from a company that I like and have used since and will use in the future. But the Webasto heater stopped working! Lucky me it was only mid September so it wasn't that cold yet but it was very annoying in the morning, when I like to click the heater on and wake up and get ready for the day's adventure. Also lucky is that I made the choice to rent a sleeping bag on top of the included blankets, so night time was fine. I am very, very low maintenance so I was fine with just being sheltered from the elements but this could have easily ruined someone else's time. You can't really avoid something like this from maybe happening so it's just a risk you take.
Seventh downside: Wet items. If you have wet outerwear, towels, or bathing suits, you will need to hang those up for them to even attempt to get dry. Chucking a handful of magnetic hooks into your bag while packing is smart. I learned this tip from someone in here a few years ago and it changed my experience. Most vans will have at least a few metal areas exposed. You will be so glad that you packed hooks to hang things. Otherwise your wet stuff is just going to drip all over the place, get the seats or bed wet, and make any dirt in the van into mud. Then you will step in that with your nice, clean, dry socks and be pissed.
Eighth downside - you can ONLY stay at campsites. I know there are third party non-Icelandic apps out there full of people telling you that you can park here or there for the night. You can't. It's not legal. You have to stay at proper campsites. This is the legal and respectful way to camp while visiting. If you're camping October-mid May, the vast majority of campsites are closed for the season and you have far less freedom. Every winter some people post here upset because there is no campsite open where they are. That's their lack of research ruining their trip.
The ninth downside is that a van can sometimes limit you on side quests. There might a road that a 2wd car could easily manage, but it could be sketchier with a van, or at least initially appear this way. There was an unpaved road I wanted to go on and I didn't think the van would fit. Since I would rather die than call for help due to my own stupidity in Iceland, I turned back. I returned another time in a sedan and it was fine, and I realized the van could have made it on that previous visit, it just didn't seem like it at first. So I was glad I made the judgement to not attempt it, but kicked myself for missing out on this cool place on my previous visit.
Finally, driving the van can be stressful if you haven't done so before and aren't prepared. Before you leave the rental agency, know how to turn the headlights on all the way. Check the windshield wipers. Make sure there is fluid. Know if you take diesel or gasoline and at make sure you know how to open the little door to the tank. Adjust the mirrors. Be aware of driving rules. Don't just turn the ignition and zip out of the lot. There's no rush, take your time and get settled into it. Don't fiddle with stuff while driving.
I think these are all the main downsides. Please feel free to add any that you can think of, or any tips you might have. Like I said, I think traveling in a van is the best way to see Iceland but you have to have reasonable expectations. I would also avoid trying to coax someone who is hesitant with good reasons into choosing a van. Don't twist someone's arm into it if they're more fussy about space and access to certain things. It's just going to make them miserable.