Gmail – Error “Web login required.” at Gmail when using an IMAP client but I’ve changed the settings to allow using it

First: Be sure to specify in your email client account settings the proper address, protocol, port and encryption for the connection, as instructed here: http://email.about.com/od/accessinggmail/f/Gmail_POP3_Settings.htm

Second: For what you want to achieve (exporting your list of contacts) google takeout seems to be the best fitting solution. More info: https://www.google.com/takeout/

(In particular, for contacts: https://www.google.com/takeout/#custom:contacts)

You are correct that the issue is because you have multiple users on that account. You're running into Google's IMAP limits.

From: https://support.google.com/mail/answer/97150?hl=en

If you have multiple users frequently accessing the same account or more than one IMAP client accessing Gmail at the same time, you may reach a connection threshold and receive the error message, 'Too many simultaneous connections.'

Gmail currently has a limit of 15 simultaneous IMAP connections per account.

IMAP and SMTP operate over different ports/connections, which is why you're still able to send, it's a different service (and also you wouldn't have many simultaneous connections to the SMTP server since that would only be opened when sending a message.)

As a possible workaround, if some of the devices are mobile devices, you could download the Gmail app (Android / iPhone) which supports multiple accounts (both personal and Google Apps will work) and won't count against your IMAP limit.