Is there anyway to accomplish this with the drill press I currently own?
As far as I can see, no.
If not, are there alternatives to a drill press that can be used for accurate holes?
Yes. You can actually drill very accurate holes (accurate in all senses: position, diameter and squareness to the surface). Note that for accurate positioning marking out must be done accurately beforehand, and centre marks or pilot holes are advisable.
If you need to drill holes square to the surface there are various techniques and tricks that can be used, some of which are covered in a previous Q&A: How do I ensure my drill is perfectly vertical before cutting a hole in my desk?
Unfortunately drilling freehand with Forstner bits is not really recommended (some guides and manufacturers of the bits say outright not to do it) and it would be difficult to ensure good results consistently. Obviously if you need to do this using a hand-held power drill you can go ahead and try, but it's highly recommended you drill practice holes first in scraps of the MDF you'll be using. I would recommend slower speeds and light pressure, do not be tempted to rush.
These practice holes may dissuade you from trying this freehand, in which case other drill bits may begin to seem an attractive proposition, see Which type of drill bit for which jobs?
The job you have to do here is one reason I'm such a fan of swing braces and have given my opinion that they're very viable even today. An inexpensive older brace with a vintage 1" Jennings or Irwin auger will drill all the holes you'd need with little effort and are capable of great accuracy with only a little practice, and proper initial layout of course.
Or.. Is there a place that rents time on a drill press that could drill 7" in from the side of a sheet of MDF?
Yes for certain places, but obviously this is very location-specific so we can't answer that for you.