3 adapters together will work in a general sense.
Reminds of my work's abomination adapter: Eternet to DB-25, DB-25 gender changer, DB-25 to DB-9, DB-9 to USB.
Now the normal DVI-VGA adapters are not active adapters, they do not actually convert the data, they just change what wires the data goes to.
You see a DVI-I port has the data for both a DVI conenction and a VGA connection, thus the adapter is not changing the data, it is simply changing the wriing configuration. You can ONLY use one of these adapters on a DVI-I port to get a VGA connction, you cant turn a VGA connection to a DVI connection.
Now if you have an active VGA to DVI adapter, then you could connect a DVI to HDMI cable or adapter and it will work (for video only, no audio).
If you dont have anything yet the much much easier route is gettng a VGA to HDMI adapter which also allows you to inject stereo or optical audio.