Typical postdoc application process:
The application process starts roughly a year in advance. Most applications includes: a research statement, teaching statement, diversity statement, cover letter, and 3-4 letters of recommendation, one of which addresses teaching.
Most applications for postdocs (but not all) are posted on MathJobs starting in the late summer/early fall, and continue to trickle in for several months. The University of California system is one, in particular, that does not usually use MathJobs. If there's a school you are interested in applying to, and you don't see an application on MathJobs, you should check their website.
Deadlines for most applications are in November and December, but some are earlier.
The NSF Mathematical Sciences Postdoctoral Research Fellowship is open to US citizens and permanent residents. The deadline is usually mid-October (next up: October 19, 2022). This fellowship is awarded to a graduate to work with a particular person at a particular institution. Coordination with that person is part of the application process.
You should communicate with your advisor regarding letter writers. Once you have decided on who you will ask to write your letters, they should be contacted NO LATER THAN ONE MONTH IN ADVANCE, but two months or more is preferable: writing a research letter for a job applicant (postdoc or tenure track) is very time consuming and people are often asked to write many of them, which requires budgeting of time. If you have a papers/preprints available, you should send the letter writer these, along with your CV and a draft of your research statement. If you don't have your research statement or preprints, you should describe your research in a couple paragraphs, and assure your letter writers that you will have a draft of the research statement at least a couple weeks ahead of the first deadline.
You can work backward from the timelines in the above descriptions to see that you will ideally want to have a draft of your research statement by the end of September.