To avoid camping, you should book hotels and hostels at least 6 weeks in advance. Hotels are available for many stages on the GR10, but sometimes you will have to do with a hostel. Hostels are called "auberge" in French, and it is possible to walk the entire GR10 while sleeping only in hotels and hostels. A night + dinner in a hostel is about 40 Euros pp, and a night in a hotel is about 50 Euros pp but without dinner. Topically, you can also get a breakfast and a lunchpack for the day.
The quality of French hotels and hostels is generally low. Expect hotels and hostels to fail to provide basics like air conditioning and wifi, even if advertised, or it may be of such low quality that it may as well not be there at all. This is a French quality issue that is not limited to the Pyrenees region. Don't get us wrong, we love France, half of us is French and the other half has a degree in French, but quality tends to be an issue. I do my best on this site to only advertise places that meet modern expectations.
The difference between hotels and hostels is a nuance. Hostels in the Pyrenees refers more the the sharing of dorms and bathrooms than to the hostel/backpacker culture that you would find in Australia, North America, and large European cities. In addition, many hostels also offer more private, double-bed rooms.
Some hotels and hostels are available through Booking.com, but many are not. Many hotels and hostels also do not use email; you have to call them to book something and then hope they have a room available when you arrive.
If you go camping (or a hybrid of camping and hostels), you gain a lot of flexibility. If you carry camping gear, it is much easier to change your plans and walk a bit more or less in a day. Camping can be done "wild" if it isn't close to a village. You can also camp on official camping grounds with facilities or on a hostel's premise with the hostel's facilities, for about 5 Euros pp.