Welcome to the Community,
unless a device isn't listed on the HCL, it isn't supported. However, this doesn't mean that there are no workarounds. I'd suggest you take a look at TinkerTry IT @ home | Install ESXi 5.5 with Realtek 8111/8168 NIC to see whether this helps. Anyway, you may instead consider to purchase a supported (or at least a "known as working" NIC like the Intel PRO/1000 GT) network adapter to avoid issues with the installation as well as with future patches/upgrades.
André