Is there any specific reason for statics on the switches?
And no, you shouldn't need to reboot anything.
For the APs. Either change the switchport connected to them to use the new mgmt VLAN as the native. Or, change the VLAN tag for the AP itself.
So for example, if the new mgmt VLAN is 700. Either change the switchport connected to the AP to be trunk native 700 and do nothing on the AP.
If instead the native on the switchport is something else let's say 100 then you'd want to type in 700 on the APs IP config. I commonly see folks specifying a tag on the AP IP config when it's the native on the switchport and that's an invalid config and you'll see errors in dashboard.