Natbib+style alpha: getting all authors' initials

There is no natbib option to do this. You can only influence the alphabetic label produced by alpha if you modify (a copy of) alpha.bst directly.

(Note that alpha isn't even a natbib style and that natbib supports mainly numeric and author-year citations, but not alphabetic styles. It probably works well enough if you don't try to use citet or citeauthor and friends, but I thought I should mention this.)

1. Locate alpha.bst on your machine. You can do this by typing kpsewhich alpha.bst into the command line/terminal. Alternatively, obtain a copy of the file from CTAN http://mirrors.ctan.org/biblio/bibtex/base/alpha.bst

2. Copy the file to a location where TeX can find it. The document directory will do fine. See also https://texfaq.org/FAQ-inst-wlcf

3. Rename the file to alpha-all.bst, say (the license of the file only allows distribution of modified copies if they bear a different name)

4. Find FUNCTION {format.lab.names} (ll. 934-973) and replace the complete function definition with

   FUNCTION {format.lab.names}
   { 's :=
     s num.names$ 'numnames :=
     numnames #1 >
       { numnames 'namesleft :=
         #1 'nameptr :=
         ""
           { namesleft #0 > }
           { nameptr numnames =
               { s nameptr "{ff }{vv }{ll}{ jj}" format.name$ "others" =
                   { "{etalchar{+}}" *
                     #1 'et.al.char.used :=
                   }
                   { s nameptr "{v{}}{l{}}" format.name$ * }
                 if$
               }
               { s nameptr "{v{}}{l{}}" format.name$ * }
             if$
             nameptr #1 + 'nameptr :=
             namesleft #1 - 'namesleft :=
           }
         while$
       }
       { s #1 "{v{}}{l{}}" format.name$
         duplicate$ text.length$ #2 <
           { pop$ s #1 "{ll}" format.name$ #3 text.prefix$ }
           'skip$
         if$
       }
     if$
   }
   

5. Add a comment with your name, the current date and a short description of the changes to the top of the file.

6. Use bibliographystyle{alpha-all} instead of bibliographystyle{alpha} in your document.

As alternative for steps 1 to 5 you can obtain the patched version of the file at https://gist.github.com/moewew/372397ed6d91f516ba71eec0097a376a

The diff between alpha.bst and the new alpha-all.bst should look roughly like this

--- alpha.bst   2010-12-09 04:18:56.000000000 +0100
+++ alpha-all.bst   2021-02-16 17:49:17.801451700 +0100
@@ -1,3 +1,14 @@
+%%%%% alpha-all.bst
+%%%%%
+%%%%% for https://tex.stackexchange.com/q/583586/35864
+%%%%%
+%%%%% 2021-02-16 MW
+%%%%%
+%%%%% modification of alpha.bst
+%%%%% that displays all author letters in the alphabetic label
+%%%%% and does not truncate the list with etalchar{+}
+%%%%% the only change is a slight simplification of format.lab.names
+%%%%%
 % BibTeX standard bibliography style `alpha'
    % Version 0.99b (8-Dec-10 release) for BibTeX versions 0.99a or later.
    % Copyright (C) 1984, 1985, 1988, 2010 Howard Trickey and Oren Patashnik.
@@ -935,10 +946,7 @@
 { 's :=
   s num.names$ 'numnames :=
   numnames #1 >
-    { numnames #4 >
-        { #3 'namesleft := }
-        { numnames 'namesleft := }
-      if$
+    { numnames 'namesleft :=
       #1 'nameptr :=
       ""
         { namesleft #0 > }
@@ -956,12 +964,6 @@
           namesleft #1 - 'namesleft :=
         }
       while$
-      numnames #4 >
-        { "{etalchar{+}}" *
-          #1 'et.al.char.used :=
-        }
-        'skip$
-      if$
     }
     { s #1 "{v{}}{l{}}" format.name$
       duplicate$ text.length$ #2 <

Now

documentclass[british]{article}
usepackage[T1]{fontenc}
usepackage{babel}

usepackage[numbers,round,longnamesfirst]{natbib}

begin{filecontents}{jobname.bib}
@book{elk,
  author    = {Anne Elk and Annie Hacker and Humphrey Appleby and Bernard Woolley and James Hacker},
  title     = {A Theory on Brontosauruses},
  year      = {1972},
  publisher = {Monthy & Co.},
  location  = {London},
}
end{filecontents}


begin{document}
Lorem citep{elk}

bibliographystyle{alpha-all}
bibliography{jobname}
end{document}

Produces

---

Here's how you would do the same thing with biblatex, which does have an option for this (maxalphanames).

documentclass[british]{article}
usepackage[T1]{fontenc}
usepackage{babel}

usepackage[backend=biber, style=alphabetic, maxbibnames=999, maxalphanames=999]{biblatex}

begin{filecontents}{jobname.bib}
@book{elk,
  author    = {Anne Elk and Annie Hacker and Humphrey Appleby and Bernard Woolley and James Hacker},
  title     = {A Theory on Brontosauruses},
  year      = {1972},
  publisher = {Monthy & Co.},
  location  = {London},
}
end{filecontents}
addbibresource{jobname.bib}

begin{document}
Lorem autocite{elk}

printbibliography
end{document}