Welcome aboard Mrsnyums, it appears you may be trying to oversimplify a fairly complex issue.
The most straightforward answer, if its used online, 72 DPI is what is needed. For printing, "it varies." Printing on what?
bitmapped artwork (artwork being images, graphics, etc.) is resolution dependent; that is, a different DPI is required for the device you are using to get the optimum results.
Without knowledge of the device, a good target is 300 DPI at the size you intend to print it.
That is important because if you take a 300 DPI 4 X 5 image and enlarge it to 8 X 10, it is now a 150 DPI image.
Which . . . may or may not be "ok." On small home printers and most laser devices, it will likely be OK. For high resolution commercial printing, definitely not OK. This is why it's device-dependent.
Line art is a whole different animal, you would generally require 600 DPI bitmapped mode files for crisp line art.
Enter vectored artwork - which is device-independent.
Vectored art is not restricted to the grids of pixels inherent in bitmapped artwork because it is rendered at the device at run time. Render vectored art to a monitor, it will render at 72 DPI. Render the same piece of art to a 2400 DPI imagesetter - you got it, it will render at 2400 DPI.
"So why not make all artwork vectored?" The nature of graphics and images make this difficult, if not impossible, as they are rendered by millions of shades of color assigned to pixels in bitmaps, which is far more efficient. There was a software we were experimenting with in the late 90's (forget it's name) that did this, allowing us to take an 8 X 10 and render it as vectored art, then reproduce it at poster size - but the results were limited and it was horrendously time intensive and expensive.
For most purposes - if you are rendering vectored art for print and web, you will need two copies - one at 72 DPI and one at print resolution, whatever that is. "Can't we take the print resolution and scale it down for web?" Yes, but you will lose some of the finer details in rescaling. It will be a far superior web image if rendered directly from the vectored program at 72 DPI.
The one item that catches my eye - and may give you problems no matter what you do -
document with vector and pixel images
This reflects that bitmapped images have been placed/embedded in the vectored artwork. If these embedded images do not have sufficient resolution to begin with, they may never be fit for printing. This is a problem with your designer, it's their responsibility to know what will be required.
There's more, but those are the cliff notes, it's not that simple of a question. :-)