She could be sick in some way, but genetics are also an issue here. I’m not sure the circumstances of how siblings were allowed to reproduce, but please do not allow this again if they are yours, or please ask the owner not to let them hatch eggs ever again.
To be blunt, or honest, there is no excuse to have unplanned chicks with birds. And related birds should never be allowed to breed. Birds lay eggs and have to be allowed to have a nest and to sit on the eggs in order for the eggs to develop and hatch. If you don’t want eggs to hatch, or in this case if it would be bad to let the eggs hatch, then the eggs should be discarded after the hen lays them, and the hen should not be allowed to build a nest or be given a nest. It is much easier to prevent unwanted eggs from hatching than to let them be incubated and hatched. If you or the owner are unwilling to throw the eggs away, then the birds should be permanently separated. It is irresponsible to allow these birds to reproduce.
As for your bird, she is probably plagued with genetic issues. There are so many in captive bred budgies, because so much inbreeding took place to get all of the colors you see. To further inbreed results in genetically weak chicks, which often die within the first few days, and otherwise die before they are a year old. Avian gastric yeast. Scaley face mites and polyoma are all diseases that are common in budgies with weak genetics. She may suffer from any or all of these, which would result in her being sick or having feather issues. She may be stunted – meaning her organs and her body have not developed properly or fully. A vet might be able to diagnose and treat some of her issues, or determine if she is in pain and suffering. Sadly she is unlikely to live for very long. If you can’t get medical care for her, the best you can do is keep her comfortable and loved for as long as she might have. I’m sorry for the little bird, but she is a sad example of what happens when related birds are allowed to reproduce. I know you feel bad for her, and watching her go through this is a hard lesson, but sadly she is the one paying the price. You should keep her in a stable environment, in her own cage, and give her plenty of individual attention. Other budgies would eventually sense her weakness and are likely to attack and kill her, so she needs to be kept alone.
Thank you for asking Lafeber,