Hi Molly,
Generally speaking, you should be able to do what you need.
To get more exact piece of advice more details are needed because the "Member not found" error says practically nothing (it is even not clear if the error is from TestComplete or from the tested application/smart card software).
The actual approach depends on how the smart card software works and how the web site interacts with it.
Once upon a time I automated something like that (it was done with TestComplete 3), but in my case the smart card software had a user interface where it was possible to enter PIN and that popped-up if connection was broken. So there was nothing special in my case: script established connection via smart card (obviously, smart card had to be in the reader) and than performed required actions. If some action failed to execute, the script verified if it was because of disconnection (and tried to reconnect then) or because of some other problem.
If your smart card software does not have user interface, then you should consult with developers and ask them how to login via API.