You can start an application of SSDI online. This will start the collection of information. Or, you can schedule an appointment at your local Social Security office. The first part of the application process includes an application for SSI, Supplemental Security Income, a needs and assets based form of welfare. The actual SSDI application usually requires that there is an expectation of full disability. You don't have to prove permanence but it should be a strong expectation.

One can earn up to $1070 per month and still apply if the limited income is based on the disability limitations.

Since you are only 3.5 months since your injury, it may be early. But, there is no harm is starting the process. At the least, you will gain an understanding of the process. The information may say the disability only needs to be expected to last 12 months but the people reviewing the application seem to ignore this part.

You said "I do not have short term disability insurance, though I was supposed to." What happened ? Do you have any recourse?