As the others have stated, they cannot shoot their heavy weapons.
The squad is a valid target for orders and would be able to carry out the order. The "immediately" simply means that you resolve the shooting before carrying on with your turn, the only change from a regular shooting attack mentioned is twin linked. The order does not instruct you to ignore any rules in any book, so moving and firing heavy weapons stands. The models would only be able to shoot thier lasguns, which probably wouldn't do anything even when twin linked.
By your interpritation since only the officer needs to have line of sight to the target vehicle/MC the ordered squad could ignore line of sight and even range to shoot at it. That would be more powerfull than moving and firing or twin-linked.
Edit: I'd also like to point out that First rank FIRE, Second rank FIRE isn't modifying rapid fire but giving a new profile for the gun. Even if it did that rule specifically states what is happening and when you can use it (lasgun), you can't give the srg a bolter and fire with the order. Fire on my Target doesn't say it changes other rules therefore it doesn't.