I have applied for jobs through the company's website and/ or sites like Indeed or craigslist jobs. Unfortunately, many times the files(PDF's that are production print quality) are too large due to the fact that they're so graphically intense. What is the best way to handle files that surpass file size limitations in an application interface?
My view is that you need to contact them and ask.
Unless you are the first person in this role they have ever hired (in which case they may not be able to judge the big files) they will have some kind of method for submitting these files, either directly (say Dropbox as stated in your question) or via flash drive etc.
If they don't, they aren't expecting files of the size you are trying to send and you should still contact them to understand their requirements (a vital skill in any job anyway).