I wish to change my greenhouse fan from a 110 unit to 220. I have the fan. The price was right. It was free. The 220 is available. Is there a way to use the 110v thermostat to power a switch (or whatever it would be called) to open and close the 220 power? If so what would this "switch" be called?
You're looking for a relay which is designed for use in Electrical Code approved installations.
First, do you want to upgrade to a safer and far more versatile 24V thermostat? The sky is the limit with those, you can get programmable, touch screen, even Internet connected thermostats like the Nest. They can also control heaters and even multi-stage. They also let you place the thermostat where you please, and do the long wire runs with easy-to-install low voltage thermostat cable.
If so, you'd want a 24V transformer to power the thermostat and remotely control the fan relay. A 40VA transformer should be plenty, and these are readily available for about $12, as they are a common heating/cooling part.
Here's a combo transformer and relay which mounts on a junction box cover. This keeps the 24V on the outside and the 120/240 on the inside, as required by Code.
Just buy one of these.
Less than $20 and they work on both 120 and 240 and have an "off" setting.