I am not sure if I am allowed to post several answers but I could not choose one as this expression can be interpreted in different ways depending on the context.

他人任せではなく自らが手を動かす -> take action; be proactive
参加者自らが手を動かす-> take part; be involved actively
管理者自らが手を動かす -> be proactive; act proactively
話を聞いて学ぶだけでなく自らが手を動かす -> put into practice; carry out in action

I think the nuance of the term "自らが手を動かす" is acting on something rather reluctant to do or opposite of being reactive, so the person needs to make some effort to do it.

"take initiative" also good, but I think the difference is whether it is something a regular person is willing to do before taking action.

P.S. Please let me know if I am not supposed to list several answers like this.