Yes, you may amphibiously assault unoccupied enemy land. You do this during the combat phase by declaring it as a battle as you would if defenders were present. The only difference is there is no dice needed to win the engagement.
Furthermore, you could invade with 1 infantry and use new transports to send in more units during Non-Combat Movement if you so wish. (For instance, perhaps you are waiting to see how the other engagements go before committing those forces to that area.)
As for straight ground combat, the same applies. You may conquer the land with any ground unit and reinforce after the engagement.
(Classic is different on reinforcements, but this is the revised board.)