Is there a way my VA can use Group Leads without Logging to my Chrome?

Yes, your virtual assistant can use it.

You need to give her the license key and have her install Group Leads Chrome extension on her chrome browser.

You also need to give her permission to write on the spreadsheet so that whenever she approves members into the group,
the data would keep getting added to the spreadsheet.

There are two ways:

1) You can click on the green share button on your google sheet and add her email address.

or

2) You can make the spreadsheet public by giving it the permission - anyone with the link can edit.

Question: Does it matter if the VA is a Moderator and not a Page Admin?

No, it doesn't matter if the virtual assistant is a moderator or page admin. 

As far as he/she can see the member requests and can see the red button on that page when logged into Group Leads.

A moderator can also use Group Leads to approve members into the group.