We've bought a couple of VigorAP 810's, and overall they look like a well thought out product - for instance, the facility for keeping the connection to iOS clients alive helped us out of a jam. However we've had to set their IP addresses manually because we can't get them to work as a DHCP client. There's a note on the bottom of the LAN setup page saying "Note: Disable "DHCP Client" or "Auto Provision" for VigorAP Management only work with Vigor2860 v3.7.3.1 and newer APM Server." I'm having a bit of trouble parsing this, but I think it's saying DHCP won't work unless you've got a Vigor router.

Is that right, or is the 810 supposed to work with a regular DHCP server?

I think if you disable the dhcp at any AP, should take automatic the dhcp from the router
You can change the dhcp range in the AP (example 192.168.1.10-100 at the router & 192.168.1.101-200 at AP) if you want to create a separate network at the AP (different password, SSID name, different channel, different keys etc)
If you have same SSID keys & network name, just disable the DHCP at the AP to avoid conflict if you are not expert

Anything connected to the 810, either wired or wireless, picks up a DHCP address just fine, but the 810 itself won't. That means I have to scratch around for an unused IP address when setting up the box, which is odd for what's otherwise a well thought out product. I don't know whether this is a known issue, an unknown issue or whether I'm doing something wrong.

(I should probably have posted this in the installation and setpup forum.)

This should just 'work' - it's not a known issue. Ask support for help.