I want to say that it's really commendable and thoughtful of you to want to help this little girl.
Forget whatever her mom tells you about school, moving to second grade, or anything, really. The stuff about advancing next year will be worked out with the school and you can't help with it. It sounds like mom might see everything through her own personal kaleidoscope, which turns everything into the teacher's fault since mom believes her daughter is perfect. In years to come things that go wrong or that the daughter does herself will be the coach's fault, the friend's fault, another parent's fault; the mom will not hold her own child accountable.This does a huge and damaging disservice to the little girl, unfortunately.
But the geographic distance between you and this family means tutoring isn't realistic, which I think you know already.
What's your relationship with the family like? For instance, do you phone them or does mom phone you, to chat, or do you only talk with her at events where one of you has traveled and you're face to face? Do you e-mail with her or are you all on Facebook--? You seem unsure about their home life (in your edit) so I'm not sure how the mom would respond to this...But I would just casually ask the mom, "I saw a series of kids' books at the bookstore that made me think of Sally -- I'd like to send them to her, just something fun for her to read over the summer as she likes. Would that be OK?" Try to make that happen and don't let the mom get an inkling that this is an attempt to help the girl (because mom will get be offended, and daughter won't ever see those books). If you are not sure of reading level, consider sending books on CD if you know the girl has access to a player.
Establishing a relationship where you can occasionally send this girl something and maybe eventually contact her to ask about what she reads is one way to help her. If her mom's OK with it you could give the girl a subscription to a good kids' magazine for her birthday or the holidays (depends on her interest and her reading level -- there are nature magazines like Ranger Rick, crafty magazines for kids, poetry and short story magazines even for younger children....).
I think if you offer to "tutor" this girl, then the mom, based on what you post here, is going to get huffy. She will figure, "My child doesn't need a tutor! She's a genius! You're saying she's not!" and mom will distance herself from you. And for a young child, face to face is really the only way to tutor. It's too bad you don't live closer, but don't beat yourself up over not being able to offer more direct help. It sounds like you'd have to navigate around a touchy mom anyway. Just try to establish contact with the mom that would lead to your sending things like a magazine subscription or books to her child without it appearing out of the blue or seeming intrusive.
Do you ever visit the area where they live? If you do, and you have established an OK relationship with the mom, you might be able to offer to take the girl out for the morning or a day when you're there. Pitch it as giving mom (and dad? where is dad?) a break. Take her someplace she might not go with her parents, if they are not likely to take her to a museum or zoo for instance.