re: Can I use the client’s existing Member ID instead of Entry ID somehow?
Sorry, no the entry ID is auto assigned in increments by the WordPress database as entries are entered into Connections. The same way which page and posts ID are auto created and incremented.
Connections does support custom fields. You could add a custom field which can be used to store the existing member ID.
re: Can Import logos? — I was thinking I could add the logos into WordPress Media, then save the urls in their Access database as a field, but how would I import that?
Sorry, no, importing logos is not possible at this time. However, the URL is exported. So, if you added the member ID custom field you could potentially sync in reverse. Export the data from Connections into you access database.
Hope this helps!