The error does not originate from DosBox but from...
a) Really poor mountings
and/or
b) From what I gather in your post...a lack of permissions.

Here's for a:
If you only mount a CD-Rom (as C, nonetheless) the install program won't find a hard drive to install to...
So

1mount c c:\somedir
2mount d d:\ -t cdrom

So you have both a hard drive and a CD in DosBox, it WON'T recognize at all anything you don't mount. Actually, if those are the mountings, I'm surprised the game installed or works at all (can you save games?) in PC#1.

The only difference really between the two computers is PC#2 has it installed on the same directory as the OS.

As for b) I don't know what "it" stands for...is it DosBox? The Dos games folder? Something else?
Anyway a folder is not the same thing as a drive (c: is NOT at all the same as c:\myfolder or c:\windows). If DosBox is installed in the Windows folder...what the hell did the person installing it do? Uninstall and reinstall to Program Files (the default location) If it's the DOS games folder...you don't want it to be there, really (UAC will prevent you from modifying the folder...and this time it might even have a good reason to). Move it somewhere else. (With that in mind, DO make a folder for your DOS games, don't mount the c: root as the DosBox c drive).
I'm assuming you're under Windows since neither Linux or OS X have a single "OS" folder. If it's something else...well, your post does lack information.

Thanks for the quick reply!

To clarify, I installed Dosbox on PC#2 to the default directory of C:\Program Files. I intended to say that it was installed on the same drive as the OS (C drive), unlike PC#1, which has DOSBox installed on a separate E: drive.

Also, if I'm understanding you correctly, I have to enter in:
mount C: <some folder on my PC>

THEN
mount D: D:\ -t cdrom

?

If so, I can try that once the computer becomes available again.

I'm new to DOSbox, so I don't know exactly what I'm doing all the time. PC#1 may have a mounting command that I placed somewhere in the config file (perhaps it's why it works)

Update:
that suggestion worked!
Unfortunately the game (harpoon) has issues installing. By any strange chance have any of you tried installing harpoon 2 using DOSBox? Searching the internet for help has proven futile thus far.

(And this is happening on both computers - I guess I just didn't try going far enough on PC#1)

Frogman3 wrote:

By any strange chance have any of you tried installing harpoon 2 using DOSBox? Searching the internet for help has proven futile thus far.

Really? 😒
Sound in Harpoon 2

You are advised to keep the 60 Seconds Guide in mind.

Please note that the first colon in your examples are unnecessary. "mount c x:\whatever" is adequate; "mount c: x:\whatever" is atypical.

Likewise "-t cdrom" will always work; "t cdrom" is questionable.

Thanks for the reply! However, I had a different problem with the installation of harpoon. After typing "install", the game begins installing and I choose soundblaster, resolution, etc.

However, it says "installation is complete" and I click "continue". The window closes and I'm sent back to the command prompt with the error stating that harpoon did not install successfully. It informs me not to panic and to just run H2SETUP again. I go back to C: and run it, but all it shows is a bunch of in-game settings and options. The game will not run and the install folder is completely empty.

EDIT:
I found that if I do a minimal install, some of the files get placed in the hparoon2 folder, but the game is still unrunnable since it's trying to read files in a folder that is empty.

You are probably not mounting things correctly.

Follow the 60 Seconds Guide.