I'm a stepmother and a mom, so I understand the pressures.
It's hard to know why uncaring biological moms check in and check out with their kids. It's very normal for a teen to be confused about having been unceremoniously abandoned by not one, but two, key women in her life (bio mom, grandmother). You may think she is too little to remember, but she has to know that something was off here. It's completely normal for her to want to be loved by someone who rejected her. She has twin siblings who know the Mom well, and it's actually validating to a 13-year-old mind to have someone love her after all. If sports is the way they can be connected, that's typical.
You want this girl to think she's as much a part of your family as the other kids, but they don't all have the same parents either. So she's searching for her roots, and she's searching for acceptance.
What's missing in this picture is your husband. When the bio mom picks up her daughter, why isn't he the one telling her when he'll pick her up or when she needs to come back? In fact, why isn't he the one talking to his ex about their child? Why does it bother you so much that she is spending 3 weeks during the summer vacation with her mother? You are being ignored and your feelings are hurt, but actually your husband is the decision-maker here unless you have adopted this teen, which you don't say you have.
What is your husband's written custody and support agreement for this child? Was anything put on paper when the child was given up by the mother or when the child was being raised by the grandmother? If not, why not? What about his visitation with the twins? Is that in a court order? What about child support, either his of the twins or the bio mom's of Alexis? If this is not formalized and filed with the court, why not? Without a legal agreement, there is nothing enforceable here - and I'm sorry to tell you that you have no legal rights whatsoever.
The bio mom knows she dropped the ball here for 13 years. Whether she's had a change of heart, or whether the twins are asking questions about their sister, or whether she has some ulterior motive, I have no idea. She may be defensive or sensitive about her lack of involvement. She knows you've been doing all the heavy lifting, and she may be embarrassed about it. Maybe that's why she doesn't speak to you - she's jealous and insecure. But if there's no legal agreement in place, she doesn't have to speak to you at all (and even if there were, the deal is going to be between the ex and your husband, not you).
My suggestion is you let this play out. Either there will be a good relationship between the teen and her mom, which is good for the child, in which case you hold your head up high and support her; OR this will play itself out, softball will end, and the mom will disappoint her and she'll come crawling home devastated, in which case she does not need a stepmom saying "I told you so" or "your mom is a loser", but one who hugs her and says "I'm so sorry you're hurt, honey."
Your focus should be on your own 3 children who may be a little baffled about where this girl has gone, and to reassure them that she loves them and hasn't left them forever, AND on your husband getting involved here about a legally enforceable agreement so that the kid knows where she belongs and that everyone cares about her. If he hasn't contacted her at all while she's at her mom's, that's bad.
But it just can't be about your hurt feelings and you being ignored. I get that's what bugs you, and it would bug me too, but it will backfire on you if you emphasize this. You have to show everyone how to be an adult here, because not everyone has been responsible over the years when it comes to being consistent for these kids.
BTW if you have your real name and your real photo up on the internet, it's best not to use your children's real names in a post. Everyone can identify you, and it's not safe for the kids.