When a student can't make themselves request a recommendation

Perhaps you could begin by expressing a lot of empathy, e.g. “yeah, your past experience really sucked and that makes this especially challenging for you,” “I can see that right now you feel like this is such a big deal, you’re almost willing to forego the internship.” “This one small step is really holding you up.” Pause, listen, after each of your understanding statements. At some point, though, you may want to ask what she plans to do to tackle this issue. If she says “I don’t know?” You may ask her if she’d like your advice, or if she would like you to make any inquiries for her regarding help available on campus. Key issue is … express complete confidence that she will be able to solve this problem like all the others she has solved in her lifetime, by deciding to do it and breaking it down into achievable steps.
Unless she is so impaired by anxiety that she is completely paralyzed, the more she can tackle on her own initiative,the better. This is how she’ll build increased resilience and confidence.
Best of luck. Parenthood is really tough.