Are Pokestop rewards only received once tapped?

No, they do not need to be tapped. You can simply let them float to the top and disappear, or hit the "X" button and close it, and the items will be collected.

source: personal experience.

To elaborate further on Dragonrage's answer:
You can go to your trainer page (lower left) and press the Journal button. You will see all rewards you've gained from pokestops there, including untapped bubbles, to confirm what items you received, even when your game lost connection or crashed after spinning it and you didn't even see any rewards.