How long will this sauce last unrefrigerated?
Depends on how much water you add to the sauce. High concentrations of sugar and salt combined with low pH (acid conditions) will prevent the growth of bacteria. If you dilute out the sugar/salt/acid, you may end up with conditions that actually promote bacterial growth. There's no way to predict the shelf-life of your sauce. Vacuum bagging is not a substitute for canning/preserving and may promote growth of anaerobes (like c. botulinum)-- if you want to keep your sauce at room temp, you can consider pressure canning it to preserve it.
ketchup is shelf-stable for at least a month even when opened. the combination of the acid in the ketchup, the antibacterial properties of the honey, and the astronomical salt level in the broth cubes should certainly prevent it from spoiling for a couple of days as long as you don't subject it to high heat or prolonged direct sunlight. i'm curious, why won't you be able to refrigerate it?