There are a whole range of tissue types which do not have capillaries and so rely on either diffusion from somewhat distant capillaries, or other oxygen sources.

The first tissue type is a bit of a trick answer. Circulating blood does not rely on oxygen diffusing out of capillaries.

The next is also a bit of a trick answer. The lungs. The alveoli are covered in capillaries but the local tissues rely on this capillaries to let CO2 out and get their oxygen from the little sack of air in the alveolus.

The next was already mentioned. The cornea has no capillaries and relies on oxygen diffusing right in from the air. If you wear contact lenses that don’t allow oxygen diffusion for too long the body can react and start emergency measures to ensure the cornea gets enough oxygen. Capillaries can start to grow into the area of the cornea. Keeps the cornea from dying from lack of oxygen but having a triangle of blood vessels growing across your pupil isn’t great for your vision.

The next is probably the first tissue type the OP was really asking about. Cartilage. You will find it in joints, tendons, ears, noses, etc. cartilage does not have a direct blood supply. Capillaries wending their way through the cartilage would, I can only imagine, wreak havoc on the strength and integrity of the tissue. Most cartilage needs to manage huge forces; whether the forces are stretch and shear in tendons or shear and compression in the lining of joints. And so they survive on oxygen and nutrients diffusing in from distant capillaries outside of the tissue proper. The effect of this is to greatly limit how quickly cartilage can grow and repair itself. Take, for example a bone break. Bone is literally full of blood and heals remarkably quickly. Usually 6 weeks and you are fully back on your feet. Even with a giant crack through it. Now take a tendon just near a bone, attached to a muscle, for example. Do a little bit of fraying of this and you could easily be facing 12 weeks to six months of healing. Transect it like many bone breaks do and without surgery the tendon may never repair.

Another tissue type lots of people might not consider is the epidermis. There are no blood vessels here and again distant diffusion is needed. Of course the very outer layer of the epidermis called the stratum corneum is made up of keratinocytes which are a little bit like waterproof bricks. They are made and layered from the bottom and shed from the surface. Ablative armouring for the body. By the time the cells are fully fledged keratin armour plates they really have very little if oxygen need.