My droid x recently got damaged and the sd reader no longer works, is there some way to download from the internet to the phone's hard drive rather than the sd card?
This is going to be application dependent. For example, Dolphin Browser HD has an option in the Settings menu to specify what directory you download files to, and you can change that to the internal memory instead of the microSD card. I don't believe you can change this option in the stock Android browser.
You could also make
/sdcard/ a symbolic link to somewhere else instead of to
/mnt/sdcard/ like it is by default. That assumes the browser uses
/sdcard, however. If it uses
/mnt/sdcard/ you'd have to edit the mounting script (part of init?) to not mount the SD card, and then create
/mnt/sdcard/ as a symbolic link to somewhere else. Something like that. Note that most/all of this requires root.