ESXi comes with drivers for a lot of NICs; for most supported NICs you usually don't need to add a driver.

Some server manufacturers install hardware where the driver isn't included in the VMware standard ESXi image but usually they provide their owen modified image which includes these drivers.

Building your own ESXi installation image isn't that difficult; it can be done using PowerCLI commands. ESXi customizer PS can automate that process for the drivers on v-front and I think there is one that MAY work with your Realtek NIC. Build that modified image and do a fresh ESXi install.

For example here https://blog.monstermuffin.org/fixing-esxi-6-5-hpe-g7-servers/ is described how to build an ESXi 6.5 image that runs on HPE ProLiant G7 servers (ESXi 6.5 is unsupported on HPE G7 servers and the HPE ESXi 6.5 image crashes because a certain module isn't compatible).

To add a driver to a already installed ESXi host you'd need a working NIC to upload it or a shared datastore (i.e. SAN) which can be accessed from another host.