The 2206 is a champ. I use one in my waveform generator.
To get more current you can drive a transistor or mosfet from the 2206. You must install a resistor in series (as my diagram above depicts). Or a potentiometer, etc.
You'll get sufficient current using a TO220 or TO3 package transistor. The common 2N3055 should be good.
I can't say whether you need the capacitor or not. I see a coil is part of the LVDT. Since a coil is involved, the capacitor may create resonance and throw off its operation.
Without a capacitor the LVDT will get a sine wave riding a DC component. Can the coil operate with an average 1/2 amp of current going through it continually? If so then it may work with this simple circuit.
However if you need true AC then you may need a more complex circuit, such as an H-bridge, etc.