Can an Apple team 'member' export from Xcode for iOS distribution?

A team member cannot request distribution signing certificates, only development signing certificates. You need to ask an admin (or another person who has got access to the distribution cert from an admin) to hand you a distributions signing certificate. They can export their certificate from their keychain or Xcode (Settings->Accounts->Signing Identities) in .p12 format and send it to you.

A team member is probably meant for persons not trusted to sign code in the name of the company. This is how Apple intended the roles to work:

https://developer.apple.com/library/ios/documentation/IDEs/Conceptual/AppDistributionGuide/ManagingYourTeam/ManagingYourTeam.html#//apple_ref/doc/uid/TP40012582-CH16-SW10

Team agent A team agent is legally responsible for the team and acts as the primary contact with Apple. The team agent can invite team members and change the access level of any other team member. There’s only one team agent.

Team admin A team admin can set the privilege levels of other team members, except the team agent. Team admins manage all assets used to sign your apps, either during development or when your team is ready to distribute an app. Team admins are the only people on a team who can sign apps for distribution on nondevelopment devices. Team admins also approve signing certificate requests made by team members.

Team member A team member can sign apps during development, but only after he or she makes a request for a development signing certificate and has that request approved by a team admin.

Yes, it is possible to sign apps for distribution as a team member. You do not need to be a team admin in order to sign apps for distribution. However, it is necessary for you to have the appropriate distribution certificate (and private key) in your keychain in order to do this. (Also - your bundleid, distribution certificate and provisioning profile must be consistent.)

You will not be able to create your own distribution certificate, only admins or above can do that. So you will need to obtain the distribution cert. from someone who already has it. Have them export it from their keychain as a p12 file. When you receive that file, double click it, enter the password, and the distribution cert. and private key will be installed on your keychain.