Sort of? Most likely the dryer outlet is only rated for 30A. Yes, you can get an adapter that will let you plug the Audi compact charger into the dryer outlet. In older homes you likely have the 3-prong NEMA 10-30 outlet for the dryer, so you would need the adaptor to connect it to the NEMA 14-50 plug on the charger. However the Audi compact charger is rated at 240V/40A, while your dryer circuit is likely rated for 30A To get the maximum charging rate available from charger without setting your house on fire, you need to use 240V/50A circuit (EV chargers are specified for "continuous use", which means the circuit they are on must have an amperage 20% above the rated value--hence, the 40A charger needs to be on a 50A circuit). A 240/50 circuit uses a NEMA14-50 plug. Any charger rated at an amperage higher than that generally must be hardwired.
The "sort of" part? You can set the Audi compact charger to operate at "50%" of max (ie, this turns the Audi charger into a 240V/20A unit.) Thus, if you set the charger at "50%" you can use it on your 30A dryer circuit because it will pull a maximum amperage of only 20A (half of 40!). HOWEVER, it will charge your EV at only half the speed, compared to running it off a 50A (effectively 40) circuit. Now, it takes about 8 or 9 hours to go from 0 to 100% charge when set to 100%, so using the dryer circuit that will take abut 16 to 18 hours. The good part is that you almost NEVER charge at home from 0 to 100%. I generally charge when it drops below 50% and only charge up to the recommended 80% for daily use. So on your dryer circuit, that would be 6 or 7 hours--something you might be able to live with.
BOTTOM LINE: Do NOT try to use your dryer circuit (using an adapter) to supply the Audi Compact Charger if the charger is set to 100%. It will try to pull 40A from your 30A circuit and you will likely fry your circuit. Set the charger to "50%" so that it will only try to pull 20A.
To set the Audi compact charger to 50 percent, plug it into the wall and press and hold the Power button for several seconds (at least 5, as I recall). The Power button is dual function: short presses are for "On/Off"; the long presses are for "50%/100%".
I have used my Audi compact charger as my home charger since Nov 2019 and it works fine. My thinking about buying a different charger was that it made no sense having a perfectly good charger rotting in my frunk, when I can just unplug the Audi charger and toss it in the frunk if I think I might need it.
I would recommend getting a 240V/50A dedicated circuit installed. There may be times when charging at 20A just doesn't cut it.
Finally, congratulations on your new EV! I have a 2019 etron quattro SUV and love it. There is just a much steeper learning curve when moving from an ICE vehicle.