ARP is only required if a device needs to know the MAC address of an IP address.
If the route between the IP address of the switch to some device is via a router (which it will be if you have a management LAN) then all the traffic will go via the router, and so you'll just the ARP entries for the router and any other directly-connected management devices.
It is possible to capture packets with a monitor port on switches, but it has many issues.
But if what you actually want is to find out which devices are where, perhaps
sw2#show mac address-table
Flags: I - Internal usage VLAN
Aging time is 300 sec
Vlan Mac Address Port Type
------------ --------------------- ---------- ----------
1 00:13:19:4e:bc:1d gi22 dynamic
1 00:20:6b:6c:25:70 gi28 dynamic
2 00:24:c4:1d:00:18 gi24 dynamic
2 00:af:1f:77:92:93 0 self
will do what you want
Also, CDP will show you switch adjacency, which might be helpful for you.
show cdp neighbors
[EDIT] And additionally, as noted in comments, if you're mapping from IP addresses to switches, you can check at the gateways for the various hosts. Depending on your network, this might be one or many routers. Of course, the ARP mapping are in the routers, but the layer 2 information is in the switches. For automated collection, consider SNMP or quick-and-easy crontabs such as
ssh router show ip arp and
ssh switch show mac-address-table