Increasing manufacturing capacity is happening. BioNTech as the most important producer in Europe has brought multiple additional partners into production as well as an entire new plant, which by itself is churning out 1 billion doses per year when fully operational (they're still scaling up production there AFAIK, but have been shipping doses from there since April). Also, the CureVac vaccine is expected to be authorized in the near future, and manufacturing capacity for that one is being prepared right now, also through manufacturing partnerships within Europe.
But the thing is: while EU agencies like the EMA are involved in these processes, and also monetary grants to finance production increases have been provided in several cases, the EU has a limited number of actions available to "increase manufacturing capacity", especially in the short term (the new BioNTech plant for example was already bought in September, so it took over half a year to get it retrofitted for production). We cannot simply - like the US or UK - declare that everything produced within the EU stays within the EU, with no exceptions. Or well, technically we could, and that would certainly increase the supply in the EU in the short term, but that would have geopolitical repercussions that might in the worst case damage the EU's vaccine production in the long term (because it is at least partially dependent on pre-products sourced from outside the EU), but will for sure cause a lot of damage in international relations. The EU is effectively the world's biggest vaccine exporter in terms of doses, especially when you focus on the mRNA vaccines which have the highest efficacy and least complications and are thus the most sought-after. Limited to just that class of vaccines, the EU even is the single relevant source right now.
The epic fuck-up of the EU body that was responsible for ordering the vaccines in the first place might in the end turn out to be a bad thing for us EU citizens because of slower vaccination, but a good thing for geopolitical relations/tensions as a whole. That's because this fuck-up kind of deters the EU from taking the same isolationist "our citizens first" stance that the US took - politically such a move would now clearly be regarded as a cheap and, most importantly, illegitimate attempt to fix the failure of negotiating proper and broad supply contracts with manufacturers using brute political force, and that effectively stops the EU from taking this step, even if it would technically be a possibility. As a result, all the countries in the world without significant mRNA vaccine production capabilities on their territories at least have realistic access to a single source for their imports, as long as they negotiated supply contracts with the manufacturers early enough. This situation has for example been the key enabler for the huge success of the Israeli's highly acclaimed vaccination campaign, which was powered by vaccine supply from Europe.