In fact I tested it with almost every option and live chat is grayed out for every one. It says open hours are between 12PM and 7:30PM PDT, and it’s 4:00PM PDT right now.
Can someone help please? Thank you!
What billing issue do you have? Unauthorized payment, refund, etc?
I just tried Contact Support > WoW > Payment > Not listed here and it took a few seconds, but the Live Chat option did show up. 15 min response estimate.
https://us.battle.net/support/en/help/product/wow/166/180/channels
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I also tried the refund option and it was available for me as well.
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The chat option may become unavailable throughout the posted hours based on the volume of requests that we get. Basically, we don’t want to simply accept an endless number of folks for Live chat to have them wait an indeterminate amount of time. So when the queue gets high, it may close temporarily.
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I accidentally added game time to the wrong WoW account. I wanted to contact Blizzard ASAP because I heard time is of the essence for them to grant my request for either a refund or game time transfer. If I try again in a few hours do you think they would still do it for me, or would too much time have passed? Thank you very much.
Time is an important factor, Suade, but that is based on when you put in the request, not when we are able to respond. 
I’d recommend putting in the request as soon as you can. Let them know what happened and they should be able to sort it out for you.
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If Live Chat does not show up for you, you can always use the Ticket option. That documents your issue in a timely way so Blizz knows you immediately tried to fix it. The time part starts from when you contact them, not when they answer 
Just don’t log in to the account you put the game time on in error.
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