I am selling my site and would like the buyer to be able to use their own LinkMachine account on the site's LinkMachine installation, rather than my account.

After logging in to LinkMachine, go to Change Settings, Account Settings. On that page you will be able to change what LinkMachine account is associated with your LinkMachine installation. After entering your current password and clicking "Change Account" on this page, you will be asked whether you would like to log in using an existing LinkMachine account, or create a new one. At this point you can pass it over to the site's new owner, who can either create their own new LinkMachine account or use their existing account if they already have one.