connecting a device that supports IP address, but not subnet mask or gateway to a network with multiple vlans

The key feature that you need to support this kind of device is proxy ARP. It is enabled by default (at least on the Cisco router I just tested on) under interface configuration mode.

If you want to be doubly sure that it is enabled, you could go into interface configuration mode and turn it on, for instance:

interface vlan37 description An interface with brain-dead printers that need proxy ARP ip address 192.168.37.1 255.255.255.0 ip proxy-arp

interface vlan38 description An interface with network gear made in the last ten years ip address 192.168.38.1 255.255.255.0 no ip proxy-arp

What these devices that have no subnet mask, a badly configured subnet mask or no default gateway will generally do is to assume that all target IP addresses are local. So they will send an ARP request. If there is a Cisco router on the segment and it has proxy-arp enabled, and if it has a route in its routing table toward the requested IP address, the router will respond to this ARP request with the MAC address of the router. The device will then forward the IP datagram to the Cisco router and the router will take care of delivery from that point on.

Thanks for the explanation. I'll test this out and post my results.

They probably have a subnet mask, but it probably assumes the subnet mask based on the ancient classful IP subnet definitions.

They don't need to have a default gateway unless the packets need to travel over a layer-3 gateway, and you don't say enough about how your network is layed out to say if there are layer-3 gateways between segments.