You'll want the compressor to engage in order to get the dye circulating. If one can of refrigerant is enough, great. If not, you'll need to add more.
Be aware that if the leak is in the evaporator, you will not be able to see the dye without considerable disassembly behind the dash. In other words, the dye may not do you any good which is why I asked how you were going to determine where the leak was. A shop that does A/C repair would typically use a sniffer to find a leak.
Also be aware that the shop may tell you that the accumulator needs to be replaced as part of the repair. With no pressure in the system along with the introduction of cans of auto parts store purchased refrigerant there is a very real possibility that the desiccant in the accumulator (sometimes called a receiver/dryer) is saturated. Moisture is the enemy of A/C systems. It will allow for the formation of acids that will eat the aluminum components of an A/C system from the inside out.
Prior to charging the system after the repair, the shop will hook up a vacuum pump to see if the system will hold a vacuum. That vacuum should "boil off" the moisture trapped in the accumulator but there is no way to actually tell if it was entirely successful. Because the shop has to warranty the repair, replacing the accumulator may not be an option.