My manager (one of the recommenders) said that he is working from home for several months due to coronavirus. It will not be possible for him to give the recommendation letter on the letterhead. Instead, he said that he would submit it via email. My question is, will this letter count towards my application?
In this era of networked printers, few people ever actually load letterhead paper into a printer any more. Instead, most organizations have electronic "identity" or "branding" collections that include templates for letterhead, business cards, etc.
As such, in my experience, most recipients (in the US and Europe at least) expect letters to arrive in PDF format. Many will not even allow a letter to be sent physically any more.
For most recommenders, then, using "letterhead" just means taking care to use the fancy formatting to make a good impression, just like one might wear formal clothing to make a good impression at a fancy event. In dealing with academics, however, eccentricity is to be expected, and variation from norms is often tolerated.
Bottom line: your recommender should probably be able to use letterhead electronically. If for some strange reason they can't, it shouldn't be a big deal, as long as it's still easy to tell they are a real and respectable person (e.g., emailing from an institutional address).
Yes, it will count. Since you write "manager" I assume it's someone in industry, not academia, and so your manager may not be familiar with academic letters of recommendation.
Probably the most important thing is that the letter should be a separate PDF document, and not just an email message. If the organization has an electronic letterhead, your recommender should use it. If not, then the recommender should put all the information one would find in a letterhead at the top of the letter. It should absolutely be mailed from the organization of the recommender, and not from firstname.lastname@example.org.
I have a facsimile letterhead and a scanned signature in blue that's set up as an Acrobat stamp. I can make a PDF look like a scanned paper letter more easily than I can print and scan a paper letter!
Finally, if your recommender is not an academic, here is some advice I've written about writing letters of recommendation which the recommender may find helpful. http://ksuweb.kennesaw.edu/faculty/rbrow211/recommendations/recommend_advice.html
The advice about FERPA is specific to the United States, and if your recommender is not an academic at an institution where you've studied, it does not apply at all. It probably doesn't apply for a letter of recommendation from an academic as there is a specific exemption in FERPA for such letterd.
Many universities have online systems for submitting recommendations in which the recommenders will get an email with a link to submit. In many of these they can paste the recommendation into a form or else upload a PDF. No signature needed. I usually do PDF with my letterhead template but it's not necessary.
I am not sure about this. You ask if a letter send by email without letterhead and without signature will be ok. If the system only accpects LOR via some annoying online portal, then the safe answer is no. The letter might never even get into the system.
You need to check with the place you are applying. My guess is that if they accept a LOR by email, then the letter you propose will probably be fine.