I've dealt with substantive (not crippling) social anxiety as a student from middle school through the public presentation of my masters' thesis in front of a couple hundred people, as an artist, and as a student teacher. This is less of an answer than testimony. I will meander a bit, covering my experiences, what worked for me, how I applied to my own course design, and some ideas for dealing with your situation.
I had my first panic attack in middle school when I was called up to the front of the room to do push-ups after smarting off. I was a mess (bright red, heart racing, couldn't breathe normally, couldn't stop crying, sweating, shaking.) I've luckily never had it so bad again, but even up through undergrad I have big holes in my memory of presentations, speeches, or anything that left me in the spotlight. We are stuck in a paradox. We desperately need more experience giving presentations we're probably never going to feel like giving. The biggest point I can make is that, in my experience, the situation could go roughly 3 ways:
Despite my discomfort, I perform to the best of my ability, avoid completely losing my place or getting so nervous I can't speak/read/etc. I have no strong sense of how well I've done, because every little mistake, tremor and quiver in my voice are magnified in my body. Observers may or may not notice; I have no clue. Ironically, I desperately need people to engage (read: not confront!) me about the presentation's topic and what I've said afterwards. I don't need things like, "you did well!" because I can't trust you not to protect me; I need engagement I can use as a proxy to assess how well I communicated. As I absorb the event, I learn that future presentations of this complexity or easier are something I can survive.
In my perception, I am overcome by the anxiety. I may have had to stop to compose myself, re-state something multiple times because my voice is quivering, transfer my speaking materials to an object more stable than my own hand. But, in reality I still managed to cover most of what I was there to say. I am alive, but the disaster I perceive to have happened is going to obliterate my ability to self-evaluate. When I calm down, meaningful engagement about the presentation, if it was indeed understandable, can help give me some less-biased data to stack against the first-hand experience. This doesn't really increase my long-run confidence in my ability to speak effectively, but it can help provide some small assurance that the world doesn't really care if I give a mediocre presentation.
I hit the point where I am too overcome by anxiety for any emotional/mental/physical coping mechanisms or crutches to get me back on track before I spiral out into panic. I'm going to need to leave; my body is too far outside of my own control for me to regain my composure over any reasonable timeframe with other people around.
What helped and how I applied it as an instructor
I suspect preparation varies per person, but what has helped me the most is resisting the temptation to prepare and memorize a speech. While this is how I was taught to present, attempting to relay anything verbatim (aside from quotes or anything else I will literally be reading) is perilous, because no amount of prep will help me find the words once the anxiety starts taking over. All of my time will have been spent preparing to say something a single fragile way, which is the wrong task. I will never feel confident in this task, because I know how easily it can get derailed.
Instead, my goal is to know what I need to say so thoroughly that I don't need to know exactly how I'll say it. I talk all the time with one or a few people about things I find interesting. Sometimes at great length. I'm trying to make presenting or giving a talk as similar as I can to this other thing I can do just fine.
I usually write a short paper which covers everything I want to say, and then decompose it and other source materials into a single stream of documentation to lead me through my points. Sometimes this means taking a source text I'll be talking about and filling it with Post-it notes and notecards; sometimes this is just a single printed document with block-quotes, marginalia, and directions to myself. From here, I spend a few hours trying to ad-lib the discussion from my notes without reading from the original paper. It doesn't always come together, but my goal is usually being able to ad-lib the discussion at least three times without getting stuck or missing anything. In this phase I may end up phrasing the same points dozens of ways.
When I got the chance to teach (freshmen!) I tried to apply what has helped me to my policy on presentations. I built three short/low-risk presentations into the course. The grading was quite generous. I permitted powerpoint, but discouraged it. I encouraged my students to come prepared to talk at slowly-increasing length about three different phases of their semester project without preparing a speech. The presentation counted as part of the grade for the assignment each was paired with, and the total number of points it contributed meant that the only way it penalized your assignment grade was if you didn't try; from there, presentations earned bonus on the assignment. In class, I let them know I sympathized strongly with not wanting to present, explained how the grading rubric was structured to give them a reason to present without the stakes being high, and assured them I was far more interested in rewarding them for talking to us about what they were working on than in penalizing them for how it went.
Addressing your problem
Others have well-covered encouraging the student to get help; I'll go in other directions. I try to shoot for policies that are flexible and anyone can take advantage of without questions. I would avoid letting the student off without doing anything unpalatable, but try to give them some agency in the tradeoff.
- Offer to hear and give feedback on presentations for all students during office hours, or at some other time before each presentation. You might even make them to talk to you about what all they plan to cover before they give it a go. Consider requiring all students who want any alternative accommodation you make available to take you up on this.
- Offer full credit for presenting to the class, X% (85?) credit for presenting to you and a smaller group of students, and Y% (70?) for presenting to you one-on-one. The group option can either be a pre-registered group of the like-minded, or can be folded into a group activity day that requires everyone to practice their presentations, while some students opt to let you grade them then.