Citizen Printer

How do I print on three part listing paper without a dot matrix printer?

Ink jet will only print the top sheet and my dot matrix has failed. Do I need to buy a new one?

Dot Matrix is the only style printer that "mechanically impacts" the paper ( thereby activating the carbon-paper like copies). Replace it with such or change your printout to have the inkjet print additional copies on single sheets.
It is either the document file is corrupted or both of your printer is bad. Try to print the document file from a different computer and printer. Before you get a new Ink Jet printer, make sure it is not the Ink Jet cartridge causes you this issue. If the sheet came out fast after printed the top part of the sheet, then it could be the Ink Jet printer is bad. But if the sheet came out normally, then the Ink Jet is fine but the cartridge is bad.
you cant. Those multiple copy forms are for impact printing - the strike leaves the imprint behind the first copy. Current printers don't impact - they melt the toner onto the page or they shoot ink in patterns. Yeah, I know, an entire industry is gone. LOL. Print three copies and be done with it.