@Courtney Duong, @Chris Seveney You're welcome!
Yes, it was all completely online in Dec 2019 - Feb 2020, from here in Dubai. I don't know if it's possible for individuals, but this was for my single-member LLC.
I registered the LLC first, then applied for the EIN. This took ages because they would only accept by fax the application. The IRS could tell me the EIN number a week later by telephone, but I needed the official letter for the bank account application (which was waiting for it). They will only send the letter by post or by fax (that you must be standing next to), but I called them from 6 different fax machines around the city and it never worked, so I don't think that option really works. Post takes a long time here but it eventually arrived.
My registered agent doesn't offer bank accounts, mail or phone services but it only costs $49 per year so I'm happy with them. While waiting for the EIN, I had plenty of time to set up a mailing address with a service that will scan it to me and then send me hard copy if I choose for a base of $10/m (Anytime Mailbox). Also a landline phone number that I can direct to my own mobile overseas for $10/m (OpenPhone).
(I'm not trying to promote any of these companies, but it took a long time to find affordable options that would really work for an overseas investor so I thought it might be helpful for others trying to do the same.)
So after a lot of preliminary admin work and transferring my LLC's initial capital contribution, I'm finally ready to go! But then... global pandemic... Ah, well. There may be some interesting investment opportunities in due course.