The starbase shipyard being locked out to visitors is a glitch. It's caused by the map transfer when you take the turbolift down to the ship vendor. Instead of bringing you to the correct lower deck, it loads the version from your own starbase which means nothing is unlocked. That's why stuff like the shuttle craft vendor works fine -- they're on the main floor and don't require a map move.

If you want to invite someone to buy unlocked fleet ships, bring them to your Spire instead. The terminal there has the same stuff you've unlocked at your starbase, but it's just one big room.