First of all, you don't need to create a NamedTemporaryFile to use make_archive; all you want is a unique filename for the make_archive file to create.
.write doesn't return a filename
To focus on that error: You are assuming that the return value of f.write is a filename you can open; just seek to the start of your file and read instead:
f.write(make_archive(f.name, 'zip', root_dir))
f.seek(0)
data = f.read()
Note that you'll also need to clean up the temporary file you created (you set delete=False):
import os
f.close()
os.unlink(f.name)
Alternatively, just omit the delete keyword to have it default to True again and only close your file afterwards, no need to unlink.
That just wrote the archive filename to a new file..
You are just writing the new archive name to your temporary file. You'd be better off just reading the archive directly:
data = open(make_archive(f.name, 'zip', root_dir), 'rb').read()
Note that now your temporary file isn't being written to at all.
Best way to do this
Avoid creating a NamedTemporaryFile altogether: Use tempfile.mkdtemp() instead, to generate a temporary directory in which to put your archive, then clean that up afterwards:
tmpdir = tempfile.mkdtemp()
try:
tmparchive = os.path.join(tmpdir, 'archive')
root_dir = "something"
data = open(make_archive(tmparchive, 'zip', root_dir), 'rb').read()
finally:
shutil.rmtree(tmpdir)