The libraries and frameworks mentioned here are good, but if all you want to do is fill in a form and flatten it, I recommend the command line tool called pdftk (PDF Toolkit).
See https://www.pdflabs.com/tools/pdftk-the-pdf-toolkit/
You can call the command line from php, and the command is
pdftk
formfile.pdf fill_form
fieldinfo.fdf output
outputfile.pdf flatten
You will need to find the format of an FDF file in order to generate the info to fill in the fields. Here's a good link for that:
http://www.tgreer.com/fdfServe.html
[Edit: The above link seems to be out of commission. Here is some more info...]
The pdftk command can generate an FDF file from a PDF form file. You can then use the generated FDF file as a sample. The form fields are the portion of the FDF file that looks like
...
<< /T(f1-1) /V(text of field) >>
<< /T(f1-2) /V(text of another field) >>
...
You might also check out php-pdftk, which is a library specific to PHP. I have not used it, but commenter Álvaro (below) recommends it.