Would PRP for dark circles affect my vision? I am a pilot, and am scared about its side effects, if any. (photo)

This scenario is extremely unlikely. Every medical procedure has risks, but to date, there has only been one reported case of blindness related to PRP that I have heard of. Dermal fillers tend to have a higher risk of blindness (Botulinum Toxin less so), and these types of incidents usually have to do with the experience and competence of the injector. Again, this is exceedingly rare, but there were 100 reports of vision loss globally due to dermal fillers in 2015* and your doctor should inform you of this very unlikely risk before the procedure.
However…. when you consider that the dermal filler market is now worth $37b a year and the demand for these procedures has increased by 200% since 2000, and the millions and millions of fillers injected every year, the chance of this happening is extremely remote.
Nevertheless, it pays to do your due diligence – and this is why an experienced injector is so valuable. Also, remember the impact of not treating your dark circles. Dark circles can lead to a loss of self-confidence.
PRP uses the body’s own growth factors (drawn from your blood and spun through a centrifuge), to help promote healing and rejuvenation. PRP also encourages collagen production, which may assist in plumping out dark hollows. Laser and micro-needling are other possible options
Disclaimer: All results are individual. All procedures have risks. Speak to your doctor about your symptoms.

Any injection under the eye could cause a clot in a vein or artery that goes to the eye. 
There are different methods of treating under the eye that are safer. Using a dermal pen avoids any risk because the pen only penetrates up to 2 mm and under the eye likely .5 mm would be the right depth. 
Using a laser to remove the top layer of skin and then applying PRP topically also does not carry the risk of a blood clot but intra ocular lenses must be used to shield the eye and still have access to the under eye trough.
An injection of Prp has its advantages and it is still my preferred technique for this issue but I always scan the area with an ultrasound to determine the anatomy of the patient and then I inject at the lower border of the trough and massage the Prp up into the inner and out corners rather than threading a needle into those areas. These different methods have different merits and I’ll use any combination depending on the conditions that I’m treating.
The most important thing is to see a doctor that specializes in Regenerative Medicine that knows the difference between dilute plasma and Platelet Rich Plasma. Most aesthetics clinics that I have seen use dilute plasma (1-2x baseline) which carries the same risks and none of the benefits. PRP needs to have a platelet concentration of 5-7 times baseline to achieve new collagen and blood vessel growth.