We assume the drug companies will be able to adapt the vaccine quickly to the new variants. If this happens then it creates a few problems..
One problem is having to continuously distribute a new vaccine to people indefinitely.
Another problem is only certain countries are capable of manufacturing the vaccine. So you have a kind of supply chain scarcity develop where only certain people in certain countries will be able to get one.
I've seen mixed reports about whether the current vaccine will hold up as variants appear.