Doing the route without wheels and then also trying to stay vegan... that is asking a lot. Too much in my opinion.
I did the GR10 last summer, with some jaunts up to the high ridges. It is hard walking (much harder than the Tour de Mont Blanc I did in 2007), because of all the ups and downs (would have been easier to stay up on the Haute Route). One of my main issues, as a diabetic, was getting the right food, and I found that it was hard not eat the local, sausage-cheese-and-bread-heavy fare. I could have cut down on the bread a little, but then I would have had a hard time carrying food that lasted more than two days. The food at the refuges I found to be excellent in most cases, with mostly fresh ingredients and lots of vegetables (it was France after all, where people really value their food). Relying on the mountain villages to get a full spectrum of healthy food was hit and miss; sometimes they had great stock, at other times just the very basics of packaged food. In those cases being able to hop on a bus or flag a car to get to the bigger towns below for proper re-supply was necessary. There was no way I could have walked those distances and still kept up the flow of the trail walk. The Pyrenees are not a single, straight line across the Iberian Peninsula... the foothills weave in and out and there are valleys everywhere. You'll wear yourself out trying to walk the ups and downs trying to get to a certain location. That's why I would say, if you're bent on not relying on the refuges, that using wheels is probably a smart move.
Last summer a huge heat wave hit the entire Pyrenees area. And this on the French side, which is usually considered the cooler, "wet" side. Some days the temperature got up to 42ºC. It is a lot hotter and drier on the Spanish side. If you are not used to high heat and how to deal with it, it can bring your walk to a halt. I know I wasn't prepared for it, even though I am used to walking in the high humidity/ heat of Japan. I hadn't realized just how fast high dry heat will suck the water out of you. My trip slowed down a lot because of needing to acclimatize and keep from getting heat stroke.
Tent camping works great if you can find a proper place to pitch your shelter. Up high on the Haute Route there were lots of places to camp, but on the ups and downs of the GR10 there were often places where all day I couldn't find a flat surface, or a lot of the land I was passing through was private land. Asking the owners to camp sometimes worked, sometimes didn't. In general I found people to be friendly and accommodating.
But I also found that the refuge and d'etape social experience added a lot to the experience of the walk. I met all kinds of people who either helped dissipate the loneliness of walking alone, or else had fantastic information for the trail ahead, things I would not have known on my own. I'd definitely recommend at least some time in the refuges just to get a better idea of the area, but also to make friends.
I encountered lots of very powerful thunder and lightning storms along the way, and one afternoon on the Petit Vignale I thought I was going to die. I got caught near the summit and had to decide to walk toward the storm to get to the refuge a half hour away, or turn back and walk 5 hours down to the valley below. I had tried setting up my shelter on a grassy knoll, but the lightning strikes were far too close for comfort, and in an utter panic I ripped up all my stakes, stuffed away the shelter, and decided to hold my breath and make for refuge, being far too tired to walk all the way back down the mountain. Stepping into the refuge, after being drenched and terrified, was one of the most welcoming moments of my life. Especially with all those people who helped me get my pack off and asked if I was all right. The hot vegetable soup that followed was one of the best meals I've ever eaten. And the bed in the stillness of the dorm room was a hundred times better than lying quaking under a thin fabric with the wind howling and thunder clapping above me.
I'd say that all in all the Pyrenees are wilder than the Alps, though not as high. Towns and villages are not as frequent. Ups and downs are a lot steeper. Keep this in mind as you plan for the journey.