Can't I scan if the printer black ink cartrige is out? The alarm

Lois, unfortunately, most printers will prevent you from doing activities when the ink/toner cartridges are empty. While it my be frustrating, there's usually no way around this other than to replaced them.
You can purchase replacement ink for your Canon Pixma MX310 from most major retailers, including several online ink-related sites, such as: http://www.databazaar.com/
You'll need to purchase a Genuine Canon replacement cartridge to get the best life out of your printer. Look for a PG-30, PG-40 or a PG-50. Each of these will fit your printer. The PG-50 holds the most ink and will last longer, but is obviously a little more expensive.
Conversely, if you need the photo cartridge replaced, look for a Canon PG-31, PG-41 or a PG-51.
Hope this helps!

I'm sorry no you can't as previously described. In most cases when your outta ink it breaks communication with the pc and fails most functions. I get my ink from www.inkgrabber.com