Is it acceptable for an undergraduate researcher to ask to work on a project that will yield publishable results?
Yes, of course! There is nothing wrong with expressing your goals for the research project to your supervisor, and I think any reasonable research supervisor would appreciate that sort of honesty from their mentees.
When I originally started working with my supervisor, I expressed that two of my goals for the summer were to (1) contribute to the group in a meaningful way and (2) the possibility of publication.
Sounds like a reasonable list of goals to me. But, for your 2nd goal, the possibility of a publication coming out of your work is only that: a "possibility." As you may be relatively new to research, it is probably a good idea for you to keep in mind that not all research projects result in publishable results (as this question highlights).
Would it be too forward of me to ask to contribute to something that will result in publication (considering that I am an undergraduate and given that there is only one month left)?
This is where it gets a bit trickier. You only have one month left on the project, and it may be difficult for your supervisor to find something meaningful, and publishable, to work on in that relatively short amount of time.
On top of that, there may be other factors working "against" you. For example, suppose that there is a gap your group is looking to fill, but you do not have the background necessary to make a meaningful contribution. What are they to do in this case? Drop everything they are doing to get you up to speed? I think you can see that this may be an unreasonable request.
For all I know, you are an extremely bright student who is more than capable of making a breakthrough on a tight deadline. But, if there are gaps in your knowledge, and it would take someone a nontrivial amount of effort to get you up to speed, then that could be problematic. There are other researchers in the group, and depending on the group dynamics, it may be difficult (or easy) for them to make accommodations for you on such short notice.
In the end, I think you should definitely have a chat with your supervisor, give them the lowdown on what you'd like to accomplish in your last month. Then, listen.