I have four one week old chicks and one constantly closes its right eye and isn't putting on as much weight as the others.
She is probably one of the more quieter ones, not the leader.

My parents think she is not putting on weight, I am not sure as she isn't as large as the others but I think she still is putting on weight.

The leader chick does like to peck the others to teach them that she is the boss (They are still figuring out their order as they are only a week old) so I am thinking she has been pecked in the eye and it is sore.
When we first got her she wasn't closing one eye and she is growing feathers at the same rate as the others, she just isn't as fluffy as the others which makes me think she is actually putting on weight. They came from a good breeder, have been sexed. pets question and answers,www.5d2d.com

Her eye isn't weeping, inflamed or obviously red. She has a pretty full crop at all times (not more or less then the others at least) and they are fed on a good mix (which at the moment IS a bit powdery just because of the way it has fallen from the feeder).

Is there anything that I can do or is this just nothing to worry about?

~ Oh and I know I should be taking her to the vet, so please don't say "take it to the vet" and then nothing else. My parents said that its not worth spending a lot of money on a 15$ chicken (Which breaks my heart) and that if there is something wrong there probably isn't much they can do. ~

You might have a few issues going on. One is that indeed she got pecked and cleaning her eye out with a soft rag and some warm water might help. If you have a good in with your vet, they might be willing to dispense some triple antibiotic eye ointment to you. Or some feed stores carry terrimycin (sp) eye ointment, but it will be expensive. I can get a tube at my local farm store for $18 bucks. If you can get some then treat her eye.
Another issue is that she is indeed sick. At her age that is not uncommon and you will need to get some antibiotic powder. Again through your vet or local farm supply stores carry it. It is a terrimycin (sp) based powder that you mix in the water for 5-7 days. All your chicks should be getting at least one treatment. If your "sick" chick is has caught an illness, then more then likely all the other ones have it to. As a note here, there are other antibiotics to try, but terrimycin is the easiest to get and usually clears up the issue chicks have. The packet of ab's treats a whole flock of many so you will need to convert the amount to how many chicks you have.
The other thing you can give her along with all the rest of the babies is either vitamins and electrolytes or sugar water. Vitamins and electrolytes come in a powder and you mix it in their water. You can get it at a farm store for very cheep and should probably have some on hand anyway. If you cant get the V&E's then add a little sugar to their water. This helps perk them up and gives them a little more energy to help snap them out of punkiness. I give mine sugar water or V&E's when I show to help them cope with the stress of travel. pets question and answers,www.5d2d.com
The food you should be giving them is Chick Starter Mash. I know you said your giving them a good quality feed, but you didn't say what it was. Chicks need the correct feed at this age for proper growth and nutrition. Protein and good amino acid complexes are very important to babies. Most feed stores will sell chick starter by the pound. I also think you can get it in 10 pound bags by Mana Pro. If you end up with some left over, it does not hurt sub adults or adults to have it mixed in with regular feed so you don't waste any left over.
Another thing to watch is her crop. It sounds as if she can process her feed, but I would keep an eye on it. If she develops a full crop but looses weight then she more then likely is crop bound. There is not much you can do except try a a eye dropper with water and hand water her and massage her crop into functioning correctly. This may or may not work.
When I had babies this age I never cave them grit (chicken) because they were to little to process it correctly. I think if you got small enough grit (fine sand like grit) for like a parakeet then you could try to offer some grit in a separate container. They may or may not use it. pets question and answers,www.5d2d.com
I will also say that it might be a good idea to contact your vet. It is a good plan to find the vets in your area who are familiar with chickens and are not afraid to treat them.

Raised over 200 chickens for show, egg and meat production. Owned a feed store for almost 18 years that helped hundreds of customners start chicks to have a pets.