Yes you can open your eyes in a tanning bed while wearing the provided tanning bed goggles.
The tanning bed goggles will block the harmful UV rays from the tanning bed so they don't harm your eyes and the goggles are see through.
So you can open your eyes while tanning as long as you're wearing the tanning bed goggles which you have to wear otherwise your eyes will be damaged if you do not.
Keeping your eyes closed is not enough to protect your eyes from the harmful and dangerous UV rays of a tanning bed but with the goggles it protects your eyes and you can keep your eyes open when in the tanning bed.