My personal recommendation would be for you to look into summer research opportunities instead (e.g. NSF REU programs, summer UROPs). Not only will it be easier to arrange, but you'll probably get to hang out with a bunch of other motivated math undergrads as well (never underestimate how much you can learn from good peers).
As for researching on your own, I would not advise it. It's incredibly difficult for an undergrad to figure out whether a particular problem is tractable with the tools you have available. Choosing a problem is something you'll definitely want to have an advisor for, especially when you're new to the field.
I have never tried finding a collaborator purely from online contact. I suspect it is difficult to nigh impossible as an undergrad, but someone please correct me if I'm wrong. However, as someone who is currently about 3000 miles away from my master's advisor and still working on research, let me say that while it is certainly possible to collaborate from a distance, this is only really feasible if one already has a good grasp on the problem at hand.
But I'm only a lowly master's student at the moment, so take everything I've written with a grain of salt. If you can find a collaborator online, all the more power to you. Still, I'd definitely recommend finding a summer program instead if at all possible. Good luck!