How do I join the next line when a line matches a regex for whole document in VI?

In normal mode, J (as distinct from j, which moves the cursor down one line) is used to join a line with the one directly beneath it. However, by default it adds a space to the end of the first line; to get the result you want (joining the lines without inserting an additional space), one would have to use gJ.

In order to use normal-mode commands in ex-mode (which you enter by pressing : while in normal mode), one must use the normal command. See :h normal within vim. So, to work with the next line that matches the pattern, one would use (note that by default, you have to escape the + to get it to work with vim's regex, a consequence of maintaining compatibility with the original vi's ancient regex engine):

:/^a.\+g$/normal gJ

To work on every line that matches the pattern, one would use the :global command (see :h :g within vim) like so:

:global/^a.\+g$/normal gJ

Or, more concisely:

:g/^a.\+g$/norm gJ

It's also possible to use the ex command join (see :h :join) to achieve the same thing with very slightly less typing (the ! at the end, in this case, tells join not to insert a space at the end of the first line).

:g/^a.\+g$/join!