BRUSSELS—Pfizer and BioNtech are set to displace AstraZeneca as the main suppliers of COVID-19 vaccines to the global COVAX programme at the start of 2022, a shift that shows the increasing importance of their shot for poorer states. The expected change comes with headaches for receiving countries that lack sufficient cold storage capacity to handle the Pfizer vaccine, and amid risks of a shortage of syringes needed to administer that shot. AstraZeneca is currently the most distributed vaccine by COVAX, according to data from Gavi, the vaccine alliance that co-manages the programme with the World Health Organization (WHO). The programme has so far delivered more than 600 million shots to nearly 150 countries, of which more than 220 million are AstraZeneca’s and about 160 million Pfizer’s. But in the first quarter of next year, Pfizer is set to take over, according to Gavi and WHO figures on doses assigned by …