I have a server that is hosted by Minehut (a server that hosts other servers inside of, great community) and with it, I can't edit much of my server files, including the world. So is there any way I could import a custom world without editing the server files?
A basic Google Search linked me to their store: http://store.minehut.com/
According to their 4th Tier plan, they offer a plugin for importing worlds called Scaffold.
As a general question, it is impossible to import worlds in Minecraft when you do not have access to the server files. This is simply because you need to migrate the world files into the /world folder, which you can't do if you don't have access.
No is the simple answer, unless, as another answer said, you buy tier 4 of the server plan. However, there is a way of doing it if you set up a new server from scratch. You can download a server from the minecraft website, then follow a tutorial to set it up (That's a whole new topic). When you've done that, go in to your server folder and replace the folder called 'world' with your map you want to use from your saves within your .minecraft folder (or wherever it is) make sure you call it 'world' too, or the server will get very confused. I hope this helped!