Printing to the edge of a page is done because of bleeds: a portion of the layout that extends beyond the trim size of a document after trimming. Note, "beyond" and not "to" the trim size because that will invariably result in a little white area at the edge of your document.
All printers and presses require a certain amount of space on either side of the paper to help feed it through, especially when toner still hasn't been fused and ink is still wet. Your layout is printed on a sheet larger than your layout to accommodate printer marks and bleeds, then cut down to your final trim size.
In answer to your specific question, yes, you can print something to the edge of the paper with a digital press (any press, actually), but, regardless to the printer (unless the technology has advanced) you need to extend bleeds to make it look clean. You need to contact your printer for the proper specs on margin and bleeds (they will need the bleeds long enough but not too long; every printer is different). All good printers will check your document to ensure everything is where it needs to be. If they don't, go find another printer.