It's certainly possible to qualify for either "normal" Social Security disability or SSI disability with an anxiety disorder. To qualify for "normal" disability, you need sufficient work history. This can be a relatively short period if you're young enough. To qualify for SSI disability, you need to have limited assets and income.
See here for details on anxiety disorders which may qualify.
Generally, it seems, you need some medical documentation to qualify for benefits. However, it seems like you can't see a doctor to do this. You may wish to contact a Social Security office (or call) to get their opinion on this.
You could possibly explore contacting a disability lawyer or similar professional. They may be willing to work on a contingency basis. In other words, they may help set you up with medical professionals etc., in exchange for a portion of any payment you may receive later. For example, the following is a post which seems to describe such a transaction (25 percent of award to the assisting counsel) :
Here's some basic links on applying for disability :
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