Per their website the best way to contact them about a billing situation is through help.openai.com.
Source:
If you have multiple, erroneous $5,000 charges for the API, I expect they would be able to get around to that fairly quickly, as you must be an extreme outlier—at the pace and rate OpenAI is extending API limits you must have requested increases at least four or five times over the last year or so. I don’t believe many developers have limits anywhere near that high.
The other thing you should do is reach out to your financial institution, they can often be much quicker at getting these things resolved. Occasionally (but not always) they will have access to more direct means of contacting merchants (email or phone) to verify charges.
In any event, you should try again through help.openai.com. Include in your message the dates, amounts, and transaction IDs (available from your financial institution) of the charges you believe are in error, a detailed explanation of why you believe these charges are erroneous, and the resolution you are expecting.
Be polite, businesslike, and unemotional in your message. Remember no one is deliberately targeting you with this—certainly not the individual who needs to read and respond to your message—even though it might feel that way. This appears to simply be a mistake, albeit a very big and painful one for you.
Trust that the person on the other end will be sympathetic and want to help you resolve the issue as quickly as possible.
If you do get an opportunity to speak directly with someone either via chat or email, you might consider asking them if there is a better, more direct way for you (or others in the same situation) to get help in resolving these billing issues should they happen again in the future.
Please update us as the situation progresses, you’re not alone in experiencing billing woes or in having difficulty reaching support. Others will certainly benefit if you share more details about your experiences—good or bad—and what you found useful or not while navigating their support apparatus.