Instead of flat mapping the inner list to a stream, you can also map it directly to a List, and then use a custom collector to append the element. The advantage of using a collector above using the stream, is that using a collector, you can get higher performance because the fact it uses 1 large addAll call, instead of separate small add calls like the Collectors.toList() does.
A solution based on this looks like:
List<InstitutionUserConnection> result = institutionsOfUser.stream(). .map(institutionUserConnectionService::getActiveInstitutionUserConnectionsByInstitution) .collect(Collector.of(ArrayList::new, List::addAll, (left, right) -> { left.addAll(right); return left; }));