a) 'Why are we using a recycling system?' is a present progressive tense. It is often used to describe an activity ongoing at current moment or recent and on-going (temporary) changes in a societal trend. This tense implies that at the moment of speaking, there is no certainty that the activity will continue forever. We expect it to stop at some point in future.
- why are you laughing? (activity ongoing at the time of speaking; it may stop soon)
- why is climate changing (a general trend started recently and affecting broader society; it may stop in future)
b) 'Why do we use a recycling system?' is a simple present tense. It is often used for describing regular habits, general facts, or repetitive work. This tense implies that the action/activity will continue indefinitely in future. We don't expect it to end at the moment of speaking.
- why do you laugh whenever you see that movie? (a habit; it may persist in future)
- why do we laugh? (a general fact; it will be true in future too)
In your case, it depends on the context. If you say 'why are we using a recycling system', it implies that it is a recent trend in society/community and it may be a temporary activity. We don't expect it to continue forever. We may stop recycling once 'the problem' has been solved.
If recycling has been in place for a long period of time and almost everyone has become used to it and no one expects to stop recycling in future, 'why do we use a recycling system' is appropriate.
Caution: it is possible that a recent trend may start exhibiting a pattern of repetitive cycle. In that case, we can use simple present rather than present continuous. For example, why do climate changes every million years?