Can I play a video game on an ESXI server with a graphics card?

Sort of.

If you have VT-D support, and have your video card passed through, you can

Puget systems has an example build here and while there's much too much to include in my answer, here's the salient points.

Your motherboard and processor matter. I've heard reports that regular mainstream boards may have vt-d disabled, and K series processors may not. Check your processor and motherboard. I'd boot into a linux livecd to check if your processor supports vt-d (proc/cpuinfo should have that?) and google if people have done that with the same model of server.

Puget had issues with nvidia cards so went with AMD. This may be an ESXI issue rather than an nvidia issue (nvidia cards seemed to work fine with KVM), and they did it with 5.5 This is finicky and your milage may vary.Check up the specific video card and ESXI version before buying a card, or test it with an existing one.

So, yeah, its possible, but it requires a fair bit of homework to pick the right parts, and testing to get it all working. At this point, if everything is compatible, and meets requirements then you can pass through a video card or 4 to a VM.