How would I build a data acquisition system for SiPMs?
However, I do not know how these steps are best realized in practice. What components should I buy to create such a data acquisition chain?

This is a large and open area of research, with literally hundreds of papers exploring different approaches.  It is not realistic to summarize an entire of field of research in a couple paragraphs.  Instead, I recommend you head over to Google Scholar and begin reading.  Fortunately, the literature is relatively accessible, and there are a lot of good online PPT presentations from CERN and elsewhere reviewing the topic.
But in general, detection is going to involve pulse shaping (to improve time resolution and prevent pile up), discrimination, and then a digital hardware to record the arrival time.  The exact form of that hardware is going to depend on your choice of detector, required time resolution, and how much work you put into the design (e.g. custom vs. off the shelf hardware).

For example, there are many boards available (for example these DIO cards), but I am not sure if they are sufficient or how they integrate with the other parts of the data acquisition chain.

I would probably try to find something designed for TCSPC.  Otherwise you're going to have to try and implement all of that yourself, which is going to be hard if you're not familiar with acquisition hardware.
Edit for more information:

I expect each event to cause a signal in only five to ten SiPMs, so the event rate per sensor is one order of magnitude lower, i.e. around 10² - 10³ er second

Typical SiPM dark count values at room temperature at on the order of 100,000 dark counts per mm^2.  Are you actually able to measure 100 events per second on your detectors?  Do you have access to liquid nitrogen or other cryogenic cooling?