No you should never use a tanning bed without tanning bed goggles.
If you use a tanning bed without use of tanning bed goggles you can slowly damage your eyes permanently.
Tanning beds put out intense UV rays and closing your eyes is not enough to protect your eyes from those UV rays so you will still damage your eyes as a result of not using the Goggles to protect your eyes.
Most tanning bed places will not allow you to go into the tanning bed without the use of the supplied goggles.
So in those places you must always put the tanning bed goggles on your eyes to protect your eyes before getting into the tanning bed.
You don't want to damage your eyes and cause blindness from tanning without the tanning goggles.