Presumably the mRNA makers will have ramped production enough that it won't matter by the time anyone needs a booster.
Pfizer is predicting that they will be able to produce 250 million doses month by the end of the year, so I guess they are getting there (depending on how long a period is acceptable).
Moderna (another mRNA vaccine) alone is estimating it will produce between 600M and 1B vaccines in 2021 already: https://www.google.com/amp/s/mobile.reuters.com/article/amp/...
Curevac is on track to apply for authorization for another mRNA vaccine in Q2: https://www.biopharma-reporter.com/Article/2021/03/22/CureVa...
It's not out of the question that the EU will be able to supply itself entirely with mRNA (or inactivated virus) past 2021.
The population you want to cover and how long it is allowed to take are both factors in whether production is a limit, so there isn't really one answer as to whether production rate is a factor.