We are trying to hook a 220 volt, 100 watt tanning bed. We have a 20 amp breaker and the wire recommeded at home depot was a 12/2 with a ground. Will this wire work or do we need a different size?

Well it is definitely not 100 watts. Can you tell us what it says on the unit. Either watts or amps is what we are looking for, along with the exact voltage.
Also, look at the plug on the end of the cord. See if it says anything on there.

I am inclined to think a 240v/20A circuit will be fine, but I'd like to be sure.

Well it is definitely not 100 watts. Can you tell us what it says on the unit. Either watts or amps is what we are looking for, along with the exact voltage.
Also, look at the plug on the end of the cord. See if it says anything on there.

I am inclined to think a 240v/20A circuit will be fine, but I'd like to be sure.

This is a 220v bed and I looked it up online and it said to use a 20 amp circuit breaker and 20amp #6-20R receptacle so we did and home depot said to use a 12/2 with ground will that be fine or should I get a 12/3 with ground wire.

No, you did good. 12/2 is perfect for that. Just color the white wire at the panel and receptacle with a black or red permanent marker.