The required letters are Math teacher, English teacher, Counselor, PFE which is submitted the same way as a LOR, and two additional optional from, say, JROTC Commander, or the Director of some community service organization where the candidate volunteers.
DS is a self-prep college reapplicant, and experienced the same situation - application SUBMITTED before Thanksgiving, but final required LOR (Math teacher) wasn't submitted until five weeks later, pushing the COMPLETION date to 30Dec. And according to the Applicant Handbook, packets are reviewed in the order they were completed, not submitted. I hate to think DS's packet got pushed back in line.
On the flip side, though, this year is different from last year in that Class of 2028 appointments were offered in a rolling basis starting the week of Thanksgiving, typically on Fridays. All remaining EA decisions were announced on 23Dec (I believe). Same with RA decisions, with appointments rolling out weekly and all remaining decisions - like DS's "thanks but not this time" - announced on 1Apr.
This year it appears all EA decisions were announced at once on or close to the end, 23Dec. I expect the same for RA. What this tells me is that the Board will review all applications and take the very top tier, as opposed to potentially offering an appointment only to see a "better" candidate downstream but not be able to offer an appointment as all slots had been filled. I'm not saying this actually happened, and certainly there is much more to the process than I know. I just see this year's process as different than last, and am less worried about the order of completion issue.
So back to the bck11's concern, yeah you don't want to pester and possibly irritate someone while at the same time asking them to say glowing things about you. Maybe explain to them in a non-pestering way that the letter is not just important, but required, and earlier is better as later may have an impact your chances.