I'm looking for some information about how academic conferences can or already do accommodate the needs of people with anxiety. For example, do you know of any conferences which have special measures in place to encourage people who have anxiety to attend (e.g. related to travelling to a conference, attending conferences, presenting at conferences)? Or perhaps, are there any ways that universities themselves make conference attendance more accessible for people with anxiety?
To a certain extent these can be "overcome" in practice, merely through dedication and practice. Extreme introverts can, for example, seem to be extroverts when necessary just by role playing.
I don't think you are going to find a lot of accommodation, maybe none at all. But you can and should consult a professional for help in overcoming any phobias you may have. Such phobias can limit your professional life.
However, note that in many fields (math and computer science, for example), many of the people that you might meet at such conferences, including the main speakers and field leaders, will probably be just like you but have worked to present a face to the world that is more open than they naturally feel. You aren't alone.
One of my colleagues, when asked, admitted that I'd "changed my personality" from the time we first met. I once suffered an academic setback simply for fear of speaking up when it was necessary to do so.
Since the interpretation of the question changed I'll add an answer for the new one, leaving the old one in place in case it helps an individual.
Some conferences have two simple features that might be adapted to your needs. The first is a "newcomers/first-timers session" early on the first day of the conference in which people will learn about what to expect in various kinds of sessions. This can, perhaps, be used to put people at ease with only minor changes.
The second thing is that many conferences also have student volunteers who help out with various things but are also allowed to attend any sessions without fee. Some student presenters are also volunteers, lowering the cost of attendance. But one of the uses of the volunteers is to be an assistant to any presenter who requests it, handling unforeseen issues so that the presenter can concentrate on what they want to say. This might be reassuring to some people that "bad things" are less likely to happen.
An additional advantage for the volunteer is that they, themselves, become more familiar and hopefully comfortable with what to expect at a professional conference so that later they don't have first-timer anxiety themselves.