The AM29LV040 is not in the supported list, but I think it should be compatible with AT29LV040. For reading, all 29X040 device should work on AM29LV040B. If you can not read the chip at all, then some thing wrong with your adapter or chip.
The 37VF040 is in the supported list, it should be OK for read/write.
It's better to check the adapter any problem since it is a "Simple" socket conversion. Just use a multimeter to make sure there are no open/short soldering on the socket.