As a part time volunteer, how can I ask my advisor to let me work independently, so that I can have a chance at first authorship?
In general, you should explain to your advisor that you are really interested in taking the lead on a research project, and ask if there is any way you can do that in the scope of your role in the lab.

Some suggestions:


Have this conversation sooner, rather than later. It's easier to sort out before too much work has been done on the project.
Ask "What do I have to do to..." rather than "Can I...". As in, "What would I have to do to be able to be first author on a paper coming out of this project, or another project?", not "Can I be first author on this project?" (He may say something like "I think that to submit a paper by December, we'll need to put in around 60 total hours per week of work" - and then you may decide that you can't do that, but that second authorship and a great letter of recommendation is better than no paper. Or not.)
Don't assume that your solution - working independently - is the only one. For example, your professor might suggest that multiple papers are expected to come out of the project, each with a different focus, and you can "specialize" in a certain part of the project and be first author on the paper on that part, even while collaborating on the project as a whole. Be open to other suggestions. 
Don't assume that you have the full picture. For example, this grad student may be working on the project with you for a few weeks just to get you started, or on the side while also pursuing another project. Etc.


Note that just like you have goals and want to put your time towards them as efficiently as possible, so does your supervisor. Supervising one student per project, especially when that one student is a temporary part-time student without much research experience,  may not be an efficient use of his time. (That doesn't mean he'd never agree to it; it does mean that you may have to be flexible about what kind of solutions you'll accept.)