Can I install a sink sprayer with a faucet not designed to support one?
Generally you want the sink sprayer attachment to be getting its water flow from the point after when the sink faucet has turned on the water and mixed the hot and cold to the desired temperature. That means you need a faucet that supports the sprayer attachment. 

There is another safety factor involved with being able to shut off the pressurized water source with a nice metal faucet instead of depending on some cheap plastic thing to do that job. If you attach the sprayer device ahead of the faucet then you have the condition that if it breaks or a hose bursts you get water everywhere. On the other hand if the sprayer fails when connected after the faucet you can very quickly get the water shut off. 

When you try to attach the sprayer ahead of the faucet you will either get all hot or all cold unless you installed a separate mixer valve. I suspect that by the time you got all through all of that it would cost as much or more than simply replacing the existing faucet with one that supports a sprayer attachment.