I'm basically forcing my younger sister to delete an Instagram post
Is that the only option that you have available?
I'm just tired of the negative energy she's producing
Normally you shouldn't tell your siblings what to do, but you may be justified in this case. Sharing too much personal information on social media can end up being harmful, and if your sister is being irresponsible, it might be best to force her to take things down.
I don't care about her feelings, all I care about is the negativity she is sharing about our family on the internet.
I can see your frustrations with her and you're justified if she is putting too much information on your family. I hope you don't stay angry with her because maybe she doesn't know any better.