Yes, you can do something to improve the situation. It's called leadership without rank. You don't need to be a manager to positively influence the people around you.
First, find out why motivation is low in your team. Was is always like that? If not, what has changed? Is the work dull, or too hard? Are your colleagues challenged? Do they believe their work is meaningful? Are they provided what they need to do good work?
Generally, people want to perform because it makes them feel better. They don't need incentives if the work environment allows them to be successful and have fun. You cannot "motivate" people by threatening to fire them and you cannot "fix" people. In any role though, you can help to improve the work environment so it's more motivating. If anything is missing to do the work properly, it's frustrating. Try to fix that by requesting stuff that does work properly.
You can find out which kind of work your colleagues love or despise. Someone hates crunching numbers, another one loves it. One coworker is really good at creating documentation, another one hates it. Of course it depends on the area you work in, but you will find this basically anywhere. Offer help, or ask the presentation lover to do the presentation for the hater.
Talk with them, listen. What bothers them? Lack of motivation and poor performance can come from personal issues, stress at home, pressure somewhere else. Talking to someone who cares already helps, and maybe you can actually offer advice. At least, you might get a better understanding why someone performs poorly.
You could be more efficient with more power, but do what you can with the little you have. It's also good practice for you if you aspire a leadership role. Leading without title is a challenge, but a title will not help you if you cannot lead. Sure, you can force people to do specific things, but that never leads to excellent performance. It's best if people respect you and know you really care for them (don't fake it, they'll know), then you can ask them to do things and they will follow.
I wish you success.