There are some things that we do not know about this environment and that impacts our ability to give good advice. It sounds like currently site A has a simple static default route and redistributes that static default route into EIGRP and advertises it to site B. And I am guessing that there is not any tracking or other validation that the static route is working. Can the original poster clarify that this is the case?
When there is only a single way to get to the Internet then there is little value in tracking or other validation of the static default route. But as you implement a second connection to the Internet then you do need some way to verify whether the static default routes are working. So at site A you will need to configure IP SLA or something similar to verify whether the static default route is working and if it is not working then to withdraw the static default route. Another approach to validate the default route would be to run a routing protocol over the link and to have the provider advertise a default route which would replace the static default route.
It is not clear how the original poster wants to operate the connections to the Internet. One possibility is to have each site use its own connection to Internet as primary and to fail over to the other site if there is a failure in the local Internet connection. Another possibility is to use one connection as preferred for both sites and use the second connection as a backup. My guess is that the preference would be the first approach. Can the original poster clarify this?
Assuming that the intent is for each site to use its own connection to Internet and to fail over to the other site as needed then at site B you would need to configure the new connection to the Internet, and to configure a default static route, to configure tracking and removal of the static default route, and to redistribute the static route into EIGRP. In this case both sites will have a static default route and will learn an EIGRP external default route. The local static default route will have better AD and will be used. And if the local static default route is withdrawn then the EIGRP external default route will be used which provides the failover capability.
One other aspect of achieving this is about address translation. I am assuming that site A is configured to do address translation for the address space of both sites. As you bring up the second connection to the Internet then site B will need to configure address translation for the address space of both sites.