Publishing into journals as an independent undergraduate researcher
Editors don't normally look at your "qualifications" when evaluating a paper. It is actually unprofessional. The work should stand on its own. 

You don't need degrees, or specifically advanced degrees to submit. Again, the work...

The costs are a different matter. I can't know the policies of all journals, but for some, at least (I hope it is still true), the following applies. First try to charge the funding agency of the research. Second, to charge the submitter's institution if available. Third (maybe second), charge the submitter. Fourth, if no one pays, absorb the costs internally. 

I'm especially not certain of the fourth point, but it used to be true for some reputable journals. 

My advice would be to submit the paper to the most appropriate journal, ignoring costs. But also ask your department if, in the eventuality that it is accepted, they would pay. If they say no, submit anyway. 

Deal with the issue of getting a personal bill only when it arrives. Possibly just by saying it is impossible for you to absorb the cost as an unsponsored researcher. If they then refuse to publish it, you have at least benefitted from some review of your ideas. 

Journals normally expect that researchers have some funding and that the funding includes covering page fees for the resulting research. Funding agencies benefit from being mentioned in the papers produced. But that doesn't apply to you and journals may still be able to make exceptions "for the greater good."