In this case some additional permission levels were added for functions associated with the hardcopy downloads. They will be found in the latest global.conf.dist but are probably missing from your global.conf
Ah yep, this was the case. I went through global.conf and global.conf.dist side-by-side in vim and adjusted a few things, so now I don't get that a permission denied error any more.
I do, however, get this one:
Failed to convert TeX to PDF with command 'cd /opt/webwork/courses/pierce_math125/html/tmp/hardcopy/work.Z40SAvbL && /usr/bin/pdflatex --shell-escape >pdflatex.stdout 2>pdflatex.stderr hardcopy' (exit=1 signal=0 core=0).
This is pdfeTeX, Version 3.141592-1.21a-2.2 (Web2C 7.5.4) \write18 enabled. I can't find the format file `pdflatex.fmt'!
kpathsea: Running mktexfmt pdflatex.fmt fmtutil: format directory `/home/pearson/.texmf-var/web2c' does not exist.
When I ran said command as root, it was successful, and created /home/pearson/.texmf-var/web2c. The web server is running under the user 'apache', so it naturally doesn't have permissions to create directories in my home directory ;).
The thing that puzzles me is that I have no idea where it is getting my username from; the only thing I can gather from pdflatex's man page is that one of their special environment variables might be set to output files to my home directory, but if that's set by webwork, I have no idea where. :/