What exactly is the problem with Yukinoshita Yukino being dependent on Hikigaya Hachiman? During the last several episodes of Oregairu Season 2 (Volume 10 in LN I think), Yukinoshita's sister was warning Hikigaya about this "sinister" relationship building up between him and Yukinoshita, and she said that their relationship is in no way genuine and that Yukinoshita is just depending on Hikigaya. 

My confusion is: What is this dependency problem exactly? Clearly Yukinoshita has developed feelings for Hikigaya in the last few episodes, so isn't relying on each other a symbolism of trust? What is the exact thing that Yukinoshita Haruno seem so worried about?
If you haven't noticed, your question is actually very difficult. Firstly, because OreGairu is probably the most psychologically intelligent anime ever in my view. But secondly, because there are actually so many aspects which everyone misses unless they watch and rewatch and understand symbolism that any answer is never completely correct.

Haruno is not evil nor is she bad. Instead, she's extremely strong willed, and Yukino has said countless times she wants to be like Haruno and be the upstanding member of her family. Alas, she's not... Haruno is. However, Haruno has always looked out for Yukino and wants her to be self-dependent but come back to their family at the same time.

Yukinon has always tried to be like Haruno. But by doing so, Haruno believes that she'll never be her own person --which is true-- while Haruno has always been her dependent. Yukinon wants to break out of her sister's shadow and be her own person, but she can't do that by relying on Hikigaya because their methods are so different and Haruno can see that Hikigaya is ridiculously intelligent which is why he gets results every time.

One problem with OreGairu right now is the lack of previous information. We don't have perfect access to everyone's background which makes sense in a way as we are able to build a picture of the characters as they are now. However, it also means we don't know exactly what happened to Yukinon as a child and Hayama and therefore, some references are really just take as they are. Yukinon has been bullied in the past as she has said and Hayama didn't help her, which is obvious. However, we don't have details around these instances so how they've affected Yukinon is opinion only.

Regardless, Haruno's words should not be taken at face value. Her saying she believes Yukinon is using Hikigaya isn't correct because everyone can see they care for each other and both would work in a relationship. However, she may believe that Hikigaya may influence Yukinon in a certain way that would affect her future and therefore doesn't want that.

Like I said, it's a difficult question but the main answer would be: Haruno has been like a mother to Yukinon for years and believes it's not in her best interest to start following Hikigaya and becoming dependent on him instead of her.