You say that the intern is already receiving help from a qualified professional. So I won't go into that side too much (aside from saying that perhaps she may benefit from more professional help than she is currently receiving, e.g. if she is currently receiving therapy, but has not explored medication, or vice versa).
With that in mind, what you have is a member of your team who suffers from a disabling medical condition. As with other instances of disability, your job is to make any reasonable adjustments to enable her to perform her role better, and it is not your job to make sure fewer adjustments are needed in any future job. She is certainly well aware of how her anxiety will limit her access to the job market, and without proper training, any attempts you make to improve her mental health will, in the long term, most likely be ineffective, or may even backfire.
The best you are able to do for her is being proactive in making reasonable adjustments. In most jurisdictions you are only required to make adjustments that are specifically requested and only if they are directly related to a proven disability. Seeing as her specific disability is anxiety, it seems unlikely that she will ask you, and she may or may not be formally diagnosed or have any proof of it being disabling.
If you go to her offering some suggestions, and asking if there are any other adjustments she might want that you haven't thought of (give her some time to think about that, it's often hard to think of things on the fly, especially if you're an anxious person), you can hopefully make the environment one where she's best able to perform her role, and from that (together with the clear and consistent feedback you ought to give any team member) build her confidence. It will also give her a chance to work out what adjustments she needs so that she can go into any future role knowing what adjustments to request.
I'm not sure if you're primarily in person or remote, but the following adjustments may be useful:
- Having the option of more remote work. Being around people can be anxiety-inducing, working from home in a familiar, comfy environment can therefore help some people be less anxious.
- Having her camera off in video calls. Just because her anxiety manifests in struggling to speak doesn't necessarily mean her talking is the cause of the anxiety, it's possible that knowing that other people can't see her might make speaking up a little less daunting.
- Communicating through text rather than speech wherever possible. A lot of anxious people find it much easier to get thoughts out through a keyboard than by speaking, so this could be a big one. An IM service like Slack is likely preferable over email for most conversations that would otherwise be over phone or in person.
- Being explicitly allowed to use certain fidget toys in meetings. There is a balancing act here between allowing her to work out any nervous energy and not wanting to distract other people in the meeting though, so it may be best combined with having the camera off in video calls, and/or training for all staff explaining that some people are better able to concentrate on meetings when engaging their hands and so people fidgeting are not necessarily ignoring the speaker, but may in fact be doing so to help themself concentrate (doing this in a way that is general rather than specific to this intern would obviously be preferable, as she would likely not want to feel singled out).
Lastly I would say that whatever reasonable adjustments you do make, or have already made, make sure that before she leaves she knows what those were, and that any future employer is legally required to make reasonable adjustments she requests in relation to a disability (this is true under both federal US law, and in the UK, as well as most Western jurisdictions, although I'm not sure of the specifics), and to encourage her to take advantage of that, and to make that request.
