Your LiveAgent plan does not limit the number of emails you can send or receive. The number of email accounts you can connect depends on your plan, see Does the built-in email account count towards the email account limit of my plan?
Sending limits in LiveAgent
LiveAgent applies default hourly sending limits for the built-in email account, for SMTP accounts and for emails to a single recipient. See What are the email sending limitations in LiveAgent?
Connecting email accounts
For more information on configuring email accounts, see the Email accounts configuration introduction. You can also find specific guides for connecting Gmail, Microsoft or other email accounts.