True but it's also a bit more complicated than that. The YouTube channel Wendover Productions has a great video on the subject: https://www.youtube.com/watch?v=byW1GExQB84 (skip to 5:01 for the talk about the cold chain).
TL;DW, there are other factors:
- Whether the hospital is able to maintain the vaccine at the required temperature (many, especially the smaller ones, don't have the equipment)
- This is medical technology and unproven options (like buying a random box off Alibaba like someone suggested) are unlikely to be used
- Pfizer (vaccine in TFA) has developed a shipment container that uses dry ice and lasts 10 days but it can only be opened twice a day, for less than a minute at a time, so facilities administering vaccines need to be able to accurately predict how many they'll administer in a given day
- Pfizer's box is only built for multiples of a thousand doses and only lasts 10 days so isn't well-suited to smaller areas that can't administer 1000 vaccines in 10 days