How is a large, spread out company supposed to work with the iOS Developer Program?
A $99 "company" developer membership allows the administrator of the account to set up many "developer" accounts. You should get the administrator of that account (whoever set it up originally, most likely) to create accounts for each of your developers.

You might need to share the admin account for things like provisioning and app submission, but for most of your daily work, you can have your own login that shares the developer membership.