The Aurora R4 has two PCIe x16 slots and two PCIe x1 slots.

http://downloads.dell.com/manuals/all-products/esuprt_desktop/esuprt_alienware_dsk/alienware-aurora-...

So you should not be limited to PCIe x1 network cards assuming that one or more of the PCIe x16 slot is open/free to use. You may want to consider getting a dual port or quad port NIC card; usually multi-port NIC cards will be PCIe x4 or PCIe x8 cards which you should be able to install into a PCIe x16 slot.

You could search a site such as Amazon or NewEgg using a search term such as Intel dual port NIC or Broadcom NetXtreme or Broadcom NIC. You could then search the VMware compatibility guide to see whether the network chip is supported or not. As most dual port/quad port NICs are used in servers, the chances are high that they are supported in ESXi.

Beware though that there are counterfeit NIC cards. I have encountered one VMware community post where the person bought an Intel quad port NIC that was in the ESXi Compatibility Guide (for a low price that was too good to be true), but ended up not being recognised by ESXi.