I hate Gog Galaxy but I used it for this.
I've tried updating my game.
Reinstalling my game.
Making sure the last Expedition completed.
I hate Gog Galaxy.
Making sure to save the last expedition after it was converted then exited the program.
Restarted my computer.
Fresh install.
No Man sky forced me to reboot when it locked up.
Before Gog galaxy I didn't have ONE Issue with NMS other than not being able to play multiplayer which sucks.
I hate Gog Galaxy so much it's made me appreciate steam which is the whole reason I ended up on GOG.

No matter what I do it will not let me start Expedition but I can access all other game modes.

Does anyone else have this problem and or a solution?

Unfortunately, the second expedition ended a couple of days ago. It only ran for a couple of weeks, unlike the first expedition.

Did you update to the latest version of No Man's Sky yet? There is a change about a message requiring an Internet connection before starting any new Community Missions so the below probably will not work with the latest version. Others have posted this in other forums as a solution but never noted anyone claiming that this did or did not work.

If you have not updated No Man's Sky yet there is a slight possibility that you might be able to complete this.
It would require first disconnecting from the Internet.
Need to disable automatic updating of system time and time zone.
Then reset system date/time to something a few days ago (something before May 31, 2021).
Try launching No Man's Sky and selecting a new game to see if the option for Community Mission is available.
Would need to complete the Community Mission before running No Man's Sky with an Internet connection and that probably means any other saves. Most can complete the second expedition in 10 hours or less.
Again, this may or may not work. You also should undo the date/time changes afterwards.

There is another option involving the Save Editor if the issue is only to obtain the special items.