A customer bought a computer online on Wednesday. On Thursday, the company delivered a broken computer. The customer immediately emailed them explaining they need a working computer before Sunday. When they didn't get a response, they phoned. The company only wanted to submit complaint tickets, but this was a time-sensitive matter, and even after the customer explained, the company wouldn't put them through to a manager or provide them with an email address or phone number where they could get help. The customer proceeded with contacting the company on live chat, followed up with numerous phone calls, and contacted the company on Facebook, but still received no help. The customer was eventually promised that they would be contacted by the company, but that didn't happen. The customer wants to receive a refund. If they submit consumer complaints, they may eventually get their money back, but the process will take months. Is there anything the customer can do because the company failed to deliver a new computer in time?

Yes, they are a few different options that the consumer has. They have done all that they can do with the company at this point. They should contact the Better Business Bureau and file a complaint about the business. The Better Business Bureau will investigate the complaint and contact the company. In some cases they can help get your money back too.
Also, the customer should contact their bank or credit card company depending on if they paid with a debit card or credit card. The bank and/or credit card company can open a dispute up right away. Which could get them their money back faster.
The customer should make sure to have all the information about the computer and the company when contacting these companies.

This is a complicated question. The short answer is yes, there are things the customer can do. It seems like the customer has gone above and beyond the call trying to contact the company, so they're going to have to try other channels. Even while they're doing this, they can submit the claim forms the computer company is asking for and jump through all the hoops. The company might reimburse the customer in the end.
First, if the customer bought from a vendor off Amazon.com or a similar site, they should check out the terms and conditions of purchasing and shipping. The site might be able to help the customer get a refund or replacement through the vendor. Plus, the customer can rate the seller low and leave a comment online.
Next, the customer could contact the company that shipped the computer—if it was an outside contractor, such as UPS or FedEx. The customer could try explaining what happened, on the off chance the computer was broken during delivery. They could ask if it was insured. If they're lucky, the shipping company might have a phone number or email for the computer manufacturer that they haven't tried.
Then the customer could file a complaint with their state's Better Business Bureau, who can warn others to be wary of this company.
If the customer is still not satisfied, and determined to be compensated, they could consult with an attorney. An initial consultation is usually free, and the attorney can advise whether it's worth the customer's money and time to pursue litigation.
Finally, the customer can share their experience on social media. Sometimes this evokes a quick response from companies that do not want their reputation smeared.