Theoretically, the answer is yes.
In practice though, I have not seen an adapter that goes from VGA to HDMI. Normally adapters go from a better interface (such as HDMI or DVI) to a lesser interface (such as VGA).
If you can find an adapter that goes VGA to HDMI it should work. Don't expect sound through HDMI to work though (since VGA does not output a sound signal, it cannot provide sound over HDMI).
Another thing you might want to consider, if you find such an adapter, it will probably be a bit on the expensive side ("odd" adapters tend to cost more than widely used/needed adapters). You may need to consider at that time if it is worth spending the money on the adapter to make use of what is probably an old monitor (if it only has VGA, it is probably more than 3 years old).