How do I append text to a file with vim script?

According to :help writefile():

When {flags} contains "a" then append mode is used, lines are
appended to the file: >
    :call writefile(["foo"], "event.log", "a")
    :call writefile(["bar"], "event.log", "a")

If you have a range of lines in your current buffer that you want to append to the log file, then

:[range]w >> my.log

does exactly what you want. (Well, maybe not exactly. For example, it has the side effect of making my.log the alternate file, and if you plan to use :e# or something, it may mess you up.)

If you already have the log message in a variable, then you could open up a scratch buffer, use append() or :put to add the line(s) to your scratch buffer, then :w >> my.log and close the scratch buffer.

:help special-buffers
:help :put
:help :w

Here is a complete log function. There is room for improvement: the combination of :new and :q may not restore the layout if you have split windows, and :put on an empty buffer leaves you with a blank line that you probably do not want.

function! Mylog(message, file)
  new
  setlocal buftype=nofile bufhidden=hide noswapfile nobuflisted
  put=a:message
  execute 'w >>' a:file
  q
endfun