If I apply to 5 different grad programs, will my supervisors have to
email the same letter to 5 different institutions?
Unless they write customized letters for each institution, then yes.
Or more likely, they'll have to email their letter one institution and fill out four different forms on four different web sites. At least one of those web sites will require setting up a new account with a secure password (containing at least one upper case letter, one lower case letter, one digit, and one special character, but no spaces, hyphens, asterisks, or emoji) and two-factor authentication. Four of the institutions will email instructions for submitting the letters to the email address you provide; one will assume that you've already given the submission instructions to each of your references. One of the reference web sites will require a recent version of Java; another will silently fail if an ad-blocker is active.
(I'm exaggerating, but only a little.)
Some of your references may be able to delegate the submission process to someone in HR, or to a clerical assistant, but everyone needs to agree on the precise protocol well in advance, so that you can include the correct contact information in your applications. As Buffy suggests, be sure that the actual letters are written by people who know you personally, who have the technical expertise to judge your suitability for graduate study, not by a random person in HR or by a clerical assistant.
I also agree with Buffy that letters that are customized to each institution are stronger, provided the customization is truly substantive. If there are clear significant differences in emphasis at different institutions—for example, one department that focuses on biology, versus another that focuses on manufacturing—then you should communicate those differences to your letter-writers. (That said, the vast majority of recommendation letters I read for successful graduate school and faculty applications are not customized, so customization may not be necessary.)
Finally, do not, under any circumstances, write the letter yourself. You do not know how to write a strong recommendation letter. You certainly do not know how to write strong recommendation letters in three different voices, all different from your own, which are consistent with previous letters "written" in those voices. I think even drafting the letter yourself is dangerous. If one of your references needs more information about your background (that isn't already adequately described in your statement of purpose and your CV, which of course you've already shared with them) to write a strong letter, talk to them face to face.