It doesn't matter what programs you have working with more than one user.
Many application simply weren't designed for this type of multi-user
environment while others will work just fine and not cause an issue.
Two possibilities, one, some later versions of Quicken have the option to
install either for only the user doing the install as in "This User," or for
"All Users." If Quicken was originally installed using the "This User"
option, it would block installation in any other user account or, the
version of Quicken you are using is simply too old and won't cooperate.
What version of Quicken are you using?
You might try removing it in the initial user account, then reinstalling it
and using the "All Users" option during setup. Then go to the limited user
account and see if that user can access simply by clicking the Quicken
shortcut.
Another possibility, given that you appear to have a number of users. If
you are using XP Pro, you may have set up a policy in Group Policy Editor,
assuming you are using it (It's only available in Pro not in Home Edition)
that is preventing installation in the other user's account.