This is probably a bug, but I figured I would check real quick to see if there is some way to fix before a send in report. The problem is when I am in the ship I am assigned as pilot of there is simply no enter cockpit button. I.E. cannot fly ship.

That happened to me a couple days ago. The enter cockpit option came up 30ish minutes later.

What kind of shop is it?

Do you see a square yellow button to the right of the Events button under Position that you can click to let you pilot the ship?

Yes I can see the yellow pilot ship button, but it does nothing if I click it. Also I have checked back several times, and the cockpit button is still not there.

You need to go into your inventory, click ships, then click Manage Piloted Ship.
Pick your site that you will be piloting and set player controlled :)