Do you mean contactless card only so they won't deal with an actual gift card? Or are they paper ones?
If a card, could you see if the company could issue an e-voucher instead that you could use online if they have an online shop? Customer service wise, a lot of companies tend to have quite useful teams on Twitter, and are often quite quick to respond.
If something like that doesn't work, there are some hashtags/tags I've seen in the past (noticed them trending) that people were using to tag in something like Watchdog when they are having an issue with a company. I saw quite a few not long back, Covid related, mainly to do with refunds. I noticed that companies seemed to be responding to the tweets, usually asking for the person to direct message them so they can sort it out privately, and likely show Watchdog (or whoever it was) they have responded and are dealing with it. It might have been when the show was on, but worth looking in to. Martin the money expert man was another I think people were tagging in tweets.