Yes, you can use Google Checkout as your only payment option, but it requires ShopSite version 8.1. Note that your customers could only checkout if they used Google Checkout and had a Google account if it is your only payment option.
Note that since ShopSite v6.2, PayPal has been a payment option in ShopSite. PayPal doesn't require a merchant account either, and you can turn off all other payment options and use them exclusively as well.