Cannot Unlock System Preferences After Upgraded to High Sierra

That is correct. You cannot install software without a password. Fix your Admin account to use a password. It's also more secure if your computer could be accessed without your knowledge or by unwanted persons.

You need to have an account login password and be using an Admin account. If you haven't ever set a password, then you need to do that:

Forgot Your Account Password

El Capitan Password Reset.

macOS Sierra-Reset a login password.

Change or reset the password of a macOS user account - Apple Support.

My administrator account has a password, but this post says a blank password might not work:

If you can't unlock settings in System Preferences - Apple Support

I think there is another System Preference setting that eliminates the prompt for a password each time you try to make a change when logged-in as an administrator. I do use that option to avoid having to provide a password each time I update a system preference as an administrator.

As I recall, it changed to always asking for a password when I upgraded to Sierra. I had to change it back to avoid the password prompt. Sorry, but I am not at may iMac and cannot recall where that option is located in System Preferences.

I think is this how I got System Preferences to stop prompting for a password while logged-in to my administrator account. Note that my administrator account does have a password.

• Click the System Preferences icon in the Dock.
• Click the Security & Privacy icon, and then click the General tab.
• If grayed out, click the Lock icon, type an administrator name and password, and then click Unlock.
• Click Advanced.
• Uncheck the Require an administrator password to access system preferences with lock icons check box.
• Click OK.
• Click the Close button.

If the Require an administrator password to access system preferences with lock icons was set on your system, then maybe setting a password, unchecking that option, and then removing the password may resolve your problem.