Unfortunately there are so many versions of this router (DI-624, DI-624+, DI-624M, DI-624S, ..) and revisions within each version (ugh), I just don’t know. I suspect it doesn’t since D-Link has typically NOT supported WDS except in their higher end models (e.g., DIR-655, DGL-4500). And a quick skim of the DI-624 manual makes no mention of it. But a check of the router’s administration pages would be definitive. WDS requires configuration. It’s typically disabled by default. And you must provide the WLAN MAC address of each device to the other. So if those options aren’t there, wireless repeating is not even possible, at least not using WDS.
But let’s assume for the sake of argument the DI-624 supports WDS.
WDS is not a certified Wifi protocol. Implementations vary enough that compatibility is always an issue. It's best to use the same manufacturer's equipment whenever possible. Even so, there may be times when compatibility is lost WITHIN the same manufacturer, esp. over time as new products lines are developed.
To be honest, WDS is often more trouble than it’s worth. Even if you have 100% compatibility, sometimes the manufacturer leaves out important details/limitations. For example, many WDS implementations only support WEP!
Having said all that, I would suggest avoiding WDS altogether (and obviously you MUST if the DI-624 doesn’t support it) and moving to a “universal” wireless bridge. A universal wireless bridge doesn’t rely on WDS. Instead, it establishes itself as a common wireless client to the other AP (just like any USB/PCI wireless adapter you might find on your desktop/laptop), then rebroadcasts the wireless signal as an AP (access point). As a result, it will work with ANY wireless router or AP! And has no better/worse limitations wrt wireless security than your average USB/PCI wireless adapter (i.e., you can typically use WPA/WPA2).
I’m just using the WAP3205 as an example. There are other similar universal solutions, including dd-wrt compatible routers (my favorite solution, but involves third party firmware). The WAP3205 has been on sale numerous times for ~$30 (after mail-in rebate). It support wireless N, numerous modes (AP, client bridge, repeater bridge (universal), repeater bridge (WDS)), has two LAN ports, etc. And excellent value if you can find it at the right price.
D-Link used to make a DWL-G800AP High Speed 2.4GHz (802.11g) Wireless Range Extender. Maybe you can find one on eBay. If you upgrade to Wireless N, you can get a DAP-1360 Wireless N Range Extender.