We are planning on having a new concrete patio poured in a couple of weeks and need to move our AC unit to make way for the slab. Is it possible for a layman to pump down the refrigerant and move the unit or do I need to pay someone to do this for me?
Unless you are an abnormally equipped layman, no. Even then, there may be a legal requirement (depending where you are) for this to be done by a licensed A/C service person, as regulations around refrigerants have become much more stringent over the years, trying to limit uncontrolled releases to atmosphere.
The hard part actually isn't pumping down the vacuum on the AC, the hard part is recovering the refrigerant. Auto parts stores will loan you a vaccum pump that will work on a house AC, but this won't allow you to recover the refrigerant. Recovering the refrigerant requires specialized equipment that's quite expensive. Discharging the refrigerant to the atmosphere is definitely illegal.
Seriously to move your AC unit, you don't need any fancy equipment to pump down your refrigerant. You simply need a set of gauges and the know how to trap the refrigerant (pump down) within the AC unit by closing the low and high side at the right time and shutting of power to the unit. Putting back in service is a different matter though.