Place the QR code at 100% as per supplied by the customer.
Check the resolution of the image in the Window>Info Panel and check that it's not too low, technically it should be hi-res image.
Make a PDF of high quality - and try to scan the QR code (there's apps for smart phones)
IF it scans ok - send the PDF to your client and ask them to confirm they are happy with it.
If the client is not happy with it - then you can ask them to send the details to you for the QR code.
InDesign has a built in QR code generator.
If that doesn't suffice there are a myriad of QR code generators on the internet that will generate a vector version.
On the QR code - it will depend on the density (how much info is stored) - so you may need to increase the size to get a good reading.
When I create vector versions
I often have a blank page starting at certain sizes - then increase the size by 1mm -
Giving me 5, 6, 7, 8, 9 10, 11, 12, 13, 14, 15 mm sizes - etc.
The most important thing is to ensure it scans well on multiple devices.
Ultimately your client should sign off on it.
The PNG may not work well - but saving to a different format would not do any good.