If you go with the adapter route, you'll need to be sure to get an adapter that works in the correct direction. Many of the video connection adapters work only in one direction and are not reversible.
It looks like VGA signal to HDMI signal requires an active, powered adapter.
I've linked a couple options. I'd have a bit more confidence in the Monoprice option working as expected, but depending on your setup, powering via USB may be a better option that powering with a wallwart transformer
Monoprice with wallwart
Option with USB power