Thanks to the two comments above and some google search, I finally found a way to make cite autocompletion work.
For Windows users out there, I have compiled the whole process:
1 - As Martin mentioned in one of the comments above, latex-suite requires grep to be installed on your system. You can get a Windows version here. Remember the directory of installation. If it is not available, vim just flickers when pressing F9 but doesn't open the completion window. Also check the grepprg setting in your .vimrc file which should be
set grepprg=grep\ -nH\ $*
according to the documentation of latex-suite. Just make sure to include this on a line in your .vimrc file.
2 - Once grep has been installed, you have to make it reachable via the windows environment variable PATH. (in words, grep will thereafter be able to be run with a command-line).
To do so, right click on computer -> properties -> Advanced System Settings -> Environment Variables.
2.a - If there is already a PATH variable, edit it. At the beginning of the code, add
C:/Program Files/GnuWin32/bin; which is the default installation folder containing the binary files of grep. If you installed it in another directory, choose the complete path to that directory. Use forward slash, not backslash. Do not forget the
; at the end, which will separate the grep path to the others already present in the PATH variable.
2.b - If there is no PATH variable, just create a new variable and name it PATH. Give it the value:
C:/Program Files/GnuWin32/bin which is the default installation folder containing the binary files of grep. If you installed it in another directory, choose the complete path to that directory. Use forward slash, not backslash.
Save and quit.
3- Try out if this helped things out. On my setup, I had the list of the sources popping in a split window, but when I pressed enter, the source code was not filled in. The reason was that I did not have a proper Python version. If you have the same problem, go to point 4.
4 - On my setup, I believe vim used version 3.2.3 of Python. This version is too recent and not compatible with vim. Go here and download Python 2.7.3. Install it. On my setup, it did the trick without further settings. Now I can happily use the cite and ref autocompletion.
Thank you all for your help! I hope this post clarifies the issue and provides a useful solution!